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zzooggaaddii ssaakkiiTTxxeebbii.. aaRRwweerriilliiaa ssaammyyaarrooss ((kkoossmmoossiiss)) ffiizziikkuurrii ddaa ggeeoommeettrriiuullii eelleemmeenntteebbii..
mmoocceemmuulliiaa ggeeooddeezziiuurrii ssaammuuSSaaooeebbiiss mmookkllee mmiimmooxxiillvvaa.. ggaannxxiilluulliiaa wweerrttiillTTaa mmddeebbaarreeoobbiiss
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ggaannxxiilluulliiaa kkuuTTxxzzoommiiTTii aaggeeggmmvveebbii,, nniivveelloobbiiss ZZiirriiTTaaddii ssaaxxeeoobbeebbii ddaa ffaarrTToobbeebbiiss
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w inasityvaoba
winamdebare saxelmZRvanelos gamocema ganapiroba boloniis
saganmanaTleblo sistemaze gadasvlam, sadac gaTvaliswinebulia
samsafexuriani umaRlesi ganaTleba, saxelmZRvanelo ZiriTadad
gankuTvnilia umaRlesi profesiuli ganaTlebis specialobis moswavle-
axalgazrdobisaTvis. igi srulad Seesabameba am saganSi arsebul
kurukulumebsa da silabusebs, romlebic aRiarebulia stu-s sainJinro
geodeziis departamentis mier.
rogorc naSromis dasaxelebidan (“geodezia topografiis
safuZvlebiT”) da anotaciidan Cans, aq ganxilulia topografiisa da
geodeziis kursis zogadi nawili. masSi mocemulia yvela is sakiTxebi,
romelTa codnac aucilebelia zemoT aRniSnuli studentebisaTvis.
aseTi saxiT saxelmZRvanelos Seqmna avtorebis pirveli cdaa da
amitomac, ra Tqma unda igi unaklo ar iqneba. isini madlierebiT miiReben
yvela saqmian SeniSvnas. aqve gvinda didi madloba movaxsenoT wignis
redaqtors baton murman mesxs, recenzents baton jano kekelias.
avtori aseve madlobas uxdis qalbaton naTia gognaZesa da daviT
papavas wignis gaformebisas gaweuli daxmarebisaTvis, baton irakli
mamacaSvils sasargeblo SeniSvnebisaTvis.
44
s a r C e v i
Tavi I samyaros (kosmosis) fizikuri da geometriuli elementebi 1.1. samyaros (kosmosis) fizikuri elementebi ----------------------------------------------------- 8
a) mzis sistemasa da metagalaqtikebs Soris gansxvaveba --------------------------- 91.2. samyaros geometriuli elementebi ----------------------------------------------------------------- 9
a) ca. cis gumbaTi. normaluri xiluli horizonti. WeSmariti horizontis sibrtye. cis sfero ----------------------------------------------------------------
10
b) heliocentruli da geocentruli moZRvrebis arsi --------------------------------- 11g) samyaroSi orientireba. samyaros polusebi da RerZebi.
saSuadReo xazi. zeniti da nadiri -----------------------------------------------------------
12
1.3. koordinatebi --------------------------------------------------------------------------------------------------- 13
Tavi II dedamiwis fizikuri elementebi
2.1. a) dedamiwis atmosfero ---------------------------------------------------------------------------------- 16b) hidrosfero -------------------------------------------------------------------------------------------------- 18
g) dedamiwis myari tani ----------------------------------------------------------------------------------- 18d) dedamiwis socialuri elementi ----------------------------------------------------------------- 18e) dedamiwis ekonomikuri elementi ---------------------------------------------------------------- 18v) dedamiwis fizikuri zedapiri --------------------------------------------------------------------- 19z) dedamiwis topografiuli elementebi ------------------------------------------------------- 19
Tavi III dedamiwis namdvili saxisa da odenobis Seswavlis (mcdeloba) cda. 3.1. geodeziis sagani ---------------------------------------------------------------------------------------------- 22 3.2. dedamiwis namdvili saxisa da odenobis dadgenis problemis arsi ---------- 23
a) dedamiwis saerTo saxe -------------------------------------------------------------------------------- 24b) dedamiwis, rogorc sferos, radiusis gansazRvris pirveli cda ---------- 25
3.3. cneba gradusuli gazomvebis Sesaxeb ------------------------------------------------------------ 26 a) triangulaciis gamoyenebis pirveli cdebi ---------------------------------------------- 27b) dedamiwis saerTo saxis dadgena -------------------------------------------------------------- 29
3.4. msoflio elifsoidi. referenc-elifsoidi--------------------------------------------------- 32
Tavi IV geodeziuri samuSaoebis mokle mimoxilva 4.1. ZiriTadi geodeziuri samuSaoebi ------------------------------------------------------------------ 344.2. gegmebis, rukebis, profilebis SedgenasTan dakavSirebuli zogierTi cnebebi --------------------------------------------------------------------------------------------
36
4.3. cnebebi agegmvebis Sesaxeb ------------------------------------------------------------------------------ 404.4. masStabebi --------------------------------------------------------------------------------------------------------- 41
a) ricxviTi masStabi --------------------------------------------------------------------------------------- 42b) xazviTi masStabi ----------------------------------------------------------------------------------------- 42g) ganivi anu transversaluri masStabi -------------------------------------------------------- 44
4.5. rukebisa da gegmebis klasifikacia ------------------------------------------------------------- 474.6. rukebis dayofa, aRniSvna – danomrva da nomenklatura --------------------------- 49
Tavi V wertilTa mdebareobis gansazRvris sakiTxisaTvis 5.1. orientireba ----------------------------------------------------------------------------------------------------- 53
a) mimarTulebis dadgenisaTvis ---------------------------------------------------------------------- 53b) WeSmariti orientireba ------------------------------------------------------------------------------- 54g) damokidebuleba pirdapir da Sebrunebul azimutebs Soris ----------------- 55d) wrfis azimutis qceva --------------------------------------------------------------------------------- 56e) wrfis sxvadasxva wertilSi azimutebs Soris damokidebuleba ------------ 57
55
5.2. direqciuli kuTxe ------------------------------------------------------------------------------------------- 57a) damokidebuleba wrfis sxvadasxva wertilSi azimutebsa da direqciul kuTxes Soris --------------------------------------------------------------------
58
5.3. meridianTa Seaxloebis kuTxis gamosaTvleli formula -------------------------- 595.4. rumbebi ------------------------------------------------------------------------------------------------------------- 60
a) WeSmariti rumbi ------------------------------------------------------------------------------------------- 60 b) damokidebuleba WeSmarit azimutebsa da WeSmarit rumbebs Soris ----- 61 g) direqciuli rumbi --------------------------------------------------------------------------------------- 61 d) damokidebuleba direqciul kuTxeebsa da direqciul rumbebs Soris -----------------------------------------------------------------------------------------
62
5.5. magnituri orientireba ----------------------------------------------------------------------------------- 62 a) kavSiri direqciul kuTxesa da magnitur azimutebs Soris ------------------ 63
5.6. cnobili azimutebiTa da direqciuli kuTxeebiT mimarTulebaTa Soris kuTxeebis gansazRvra ------------------------------------------------------------------------
65
5.7. cnobili direqciuli rumbebiT astrolabiuri kuTxeebis gansazRvra ----- 66 5.8. mixrilobis gansazRvra ---------------------------------------------------------------------------------- 68
Tavi VI koordinatebi sainJinro geodeziaSi 6.1. dekartis marTkuTxa koordinatebi RerZze -------------------------------------------------- 69 6.2. marTkuTxa koordinatebi sibrtyeze -------------------------------------------------------------- 69
a) marcxena sistema ----------------------------------------------------------------------------------------- 69 b) marjvena sistema ----------------------------------------------------------------------------------------- 70
6.3. marTkuTxa koordinatebi sivrceSi ----------------------------------------------------- 71 6.4. marTkuTxa koordinatebi sferoze (zoldneris koordinatebi) ------ 726.5. polaruli koordinatebi sibrtyeze ------------------------------------------------------------- 73
a) maTematikaSi ------------------------------------------------------------------------------------------------- 73b) polaruli koordinatebi sibrtyeze (geodeziaSi) ----------------------------------- 74g) polaruli koordinatebi sivrceSi ------------------------------------------------------------ 74d) polaruli koordinatebi sferoidze ------------------------------------------------------- 75
6.6. bipolaruli koordinatebi ---------------------------------------------------------------------------- 76
Tavi VII pirobiTi aRniSvnebi 7.1. gegmis Sedgena ------------------------------------------------------------------------------------------------- 777.2. niSnulebiani gegmilebi ---------------------------------------------------------------------------------- 80
a) wertili sivrceSi ---------------------------------------------------------------------------------------- 80b) xazi sivrceSi ----------------------------------------------------------------------------------------------- 81 g) graduireba -------------------------------------------------------------------------------------------------- 84d) profilis anu ujrediani – (milimetrebiani) qaRaldis xerxi -------------- 87
7.3. niSnulebiani proeqciebis ganzogadoeba ------------------------------------------------------ 88 a) normaluri kveTis simaRlis gansazRvra -------------------------------------------------- 93
Tavi VIII Tarazoebi 8.1. Tarazos Teoria ---------------------------------------------------------------------------------------------- 97
a) Tarazos mgrZnobiaroba ----------------------------------------------------------------------------- 102 b) xazebisa da sibrtyeebis horizontul mdgomareobaSi moyvana da Tarazos Sesworeba ----------------------------------------------------------------
103
g) Tarazos RerZis Sesworebis geometriuli interpretacia --------------------- 104d) Tarazos RerZis daxris kuTxis gansazRvra -------------------------------------------- 105
Tavi IX samiznebeli xelsawyoebi 9.1. Wogri ---------------------------------------------------------------------------------------------------------------- 107
a) Wogris gamadidebloba ------------------------------------------------------------------------------- 107b) ganvsazRvroT gamadidebloba okularisaTvis ----------------------------------------- 109
9.2. mxedvelobis are ---------------------------------------------------------------------------------------------- 111
66
9.3. gamadidebloba ----------------------------------------------------------------------------------------------- 112 a) Wogris samzerad dayeneba -------------------------------------------------------------------------- 113 b) Wogris optikuri Zala -------------------------------------------------------------------------------- 114
Tavi X danayofebis amTvleli xelsawyoebi 10.1. anaTvlebis aRebis meTodi --------------------------------------------------------------------------- 115
Tavi XI gazomvebi da xazebis dasarva 11.1. pirdapiri gazomvebi ---------------------------------------------------------------------------------------- 117 11.2. xazebis dasarva ---------------------------------------------------------------------------------------------- 121 11.3. xazebis dasarvis SemTxvevebi ------------------------------------------------------------------------ 123
a) vake adgilze dasmuli ori saris mixedviT xazis gagrZeleba ---------------- 123
b) xeobebis erT mxareze dasmuli ori saris mixedviT xazis gagrZeleba mis meore mxareze --------------------------------------------------------------------
124
g) xazis gagrZeleba mcire dabrkolebis gadalaxviT ---------------------------------- 12511.4. or dasmul sars Soris xazis dasarva ------------------------------------------------------- 126
a) or urTierT uxilav wertils Soris maRlobis dasarva ----------------------- 126 b) xeobis orive mxareze dasmul sarebs Soris xazis dasarva ------------------- 129
Tavi XII zogierTi cnobebi ganazomTa Secdomebis Teoriidan 12.1 gazomvebis saxeebi ------------------------------------------------------------------------------------------- 131 12.2 gazomvebis Secdomebi da Sesworebani --------------------------------------------------------- 131 12.3 ganazomTa rigebis saxeebi ----------------------------------------------------------------------------- 13412.4 ganazomTa Secdomebis saxeebi ---------------------------------------------------------------------- 13412.5 SemTxveviT SecdomaTa Tvisebebi ------------------------------------------------------------------- 13512.6. gazomvis erTeulebi ------------------------------------------------------------------------------------- 135
Tavi XIII kuTxzomiTi agegmva 13.1. xelsawyoebis Semowmebebi ------------------------------------------------------------------------------ 14213.2. Sveuli wredis nul adgili ------------------------------------------------------------------------- 14813.3. instrumentis dacentvrisa da reduqciis Secdoma ----------------------------------- 15313.4. kuTxeebis gazomva ------------------------------------------------------------------------------------------ 155
a) ileTebis xerxebi ----------------------------------------------------------------------------------------- 156 b) wriuli ileTebis xerxi ------------------------------------------------------------------------------ 158g) ganmeorebiTi xerxi -------------------------------------------------------------------------------------- 159
13.5. gazomili kuTxeebis gawonasworeba ------------------------------------------------------------- 162 13.6. direqciuli kuTxeebis gamoTvla ----------------------------------------------------------------- 163 13.7. pirdapiri da Sebrunebuli geodeziuri amocana ---------------------------------------- 164 13.8. wvliladebis gadatana ---------------------------------------------------------------------------------- 167
Tavi XIV niveloba 14.1. trigonometriuli niveloba ------------------------------------------------------------------------- 17014.2. geometriuli niveloba ---------------------------------------------------------------------------------- 17214.3. nivelirebi ------------------------------------------------------------------------------------------------------- 175
a) Semowmebebi --------------------------------------------------------------------------------------------------- 175b) Tarazos safasuris gansazRvra lartyis saSualebiT -------------------------- 176 g) xSuli nivelirebis Semowmeba ------------------------------------------------------------------- 177d) gadasadeb Wogriani nivelirebis Semowmeba ---------------------------------------------- 180
14.4. nivelobis warmoeba --------------------------------------------------------------------------------------- 181 14.5. mrudeebis dakvalva --------------------------------------------------------------------------------------- 185
a) marTobebis xerxi ----------------------------------------------------------------------------------------- 186 b) kuTxeebis xerxi ------------------------------------------------------------------------------------------- 187g) wagrZelebuli qordebis meTodi ---------------------------------------------------------------- 188 d) serpantini ----------------------------------------------------------------------------------------------------- 189
77
14.6. barometruli niveloba --------------------------------------------------------------------------------- 192 a) diferencialuri barometri ------------------------------------------------------------------------ 196 b) aneroidi ------------------------------------------------------------------------------------------------------- 197
Tavi XV farTobebis gamoTvla -------------------------------------------------------------------------------------
200
88
I Tavi
samyaros (kosmosis) fizikuri da geometriuli elementebi
1.1. samyaros (kosmosis) fizikuri elementebi
kosmosi anu samyaro ewodeba usazRvro sivrcesa da droSi
metagalaqtikebis gaerTianebaTa usasrulo simravles. yoveli
metagalaqtika galaqtikebis erTobliobaa. saerTod galaqtika ki
varskvlavTa sistemebis (ojaxebis) simravles warmoadgens.
es gaerTianebebi niutonis msoflio mizidulobis (gravitaciis)
kanons Seesabameba da nebismieri nivTierebis Semcveli materiis
obieqturobis, aqtiurobis, kanonzomierebis, mudmivobisa da Secnobadobis
uzogadesi Tvisebebis gamo erTiani, maradiuli, usasrulo, mravalferovani
materialuri samyaros saxiT unda warmovidginoT.
materialuri samyaro fardobiTi WeSmaritebis uwyvetliv
gamovlinebis gziT, absoluturi WeSmaritebisadmi usasrulod miaxloebis
obieqtia. amave dros igi materiis agebulebisa da moZraobis gigantur
laboratorias warmoadgens, sadac adgili aqvT: udides wnevebs, kolosalur
temperaturebs, uZlieres magnitur velebs, absolutur vakuums, procesebs
da elementarul nawilakebs, romlebic Tanxlebulni arian udidesi da
zeudidesi energiis gamoyofiT.
adgilobrivi saxelwodebis metagalaqtikaSi Semavali saSualo
odenobis galaqtikas, romelsac irmis naxtomis an rZis• gzis sistemis
saxelwodeba aqvs da romelSic sxva varskvlavT sistemebTan erTad mzis
ojaxic Sedis, samyaroSi metad mcire sivrce uWiravs, miuxedavad imisa, rom
masSi as miliardamde urTierT ramodenime sinaTlis wliT** daSorebuli
varskvlavebi sruliad normalurad moZraoben.
magaliTad, rom gavakeToT globusi, romelzedac dagegmilebuli
varskvlavebi warmovidginoT wvimis wveTebis saxiT, am globusze wveTebs
Soris manZili 65 km iqneba.
mzis sistemaSi Semavali obieqtebi, rogoricaa planetebi: merkuri,
venera, dedamiwa, marsi, iupiteri, saturni, urani, neptuni, plutoni Tavisi 32
•• galaqtika rZis gzis berZnuli saxelwodebaa (varskvlavebian caze, kididan kidemde garkveviT gamoyofili rZisferi naTeli farTo zoli) ** sinaTlis weli =60 ⋅ 60 ⋅ 24 ⋅ 365 ⋅ 298800 =9500000000000 =9,5 ⋅ 1012=9500 miliardi km/weliwadSi
99
TanamgzavriT (erT-erTi Tanamgzavria mTvare), uamravi mcire planetebiT,
kometebi, meteorebi sruliad normalurad moZraoben misi centraluri erTi
varskvlavis – mzis udidesi gravitaciuli mizidulobis Zalis gavleniT.
a) mzis sistemasa da metagalaqtikebs
Soris gansxvaveba
arsebobs ori ZiriTadi gansxvaveba:
I – mzis sistemas centraluri sxeuli
gaaCnia (es aris erT-erTi varskvlavi – mze).
galaqtikebs da metagalaqtikebs centraluri
sxeuli ar gaaCniaT – maTi centria varskvlavT
simravlis saerTo masaTa centri.
II – gansxvaveba mdgomareobs imaSi, rom
WogriT mzeris dros mzis sistemis obieqtebis diskoebi gadidebulni Canan,
maSin rodesac galaqtikis varskvlavebi damzeris dros ar
diddebian, imatebs maTi mxolod sikaSkaSe, brwyinvaleba.
varskvlavT samyaros Seswavlis mizniT varskvlavebs
yofen did jgufebad, romelTac Tanavarskvlavedebi ewodeba. am
Tanavarskvlavedebs arqmevdnen cxovelebis, gmiri adamianebis
saxelebs. magaliTad: didi daTvi, mcire daTvi, didi ZaRli da
sxva.
TviT TanavarskvlavedSi Semaval varskvlavebs maTi sikaSkaSis
mixedviT arqmeven berZnuli alfabetis sawyis asoebs.
1.2. samyaros geometriuli elementebi
samyaros geometriuli elementebis codna saWiroa imisaTvis, rom
SevZloT: 1) adgilis nebismieri wertilis mdebareobis (koordinatebis)
gansazRvra, 2) kosmosuri elementebis sivrcobrivi ganlagebis gansazRvra, 3)
mzisadmi dedamiwis qcevebis gansazRvra (dedamiwa ganicdis oTxi saxis
moZraobas).
1100
a) ca. cis gumbaTi. normaluri xiluli horizonti. WeSmariti
horizontis sibrtye. cis sfero
ca. geometriuli warmodgeniT adamiani cas uwodebs dedamiwis
atmosferos Semcvel naxevrad gamWvirvale fardas, romelic gars akravs
dedamiwas da romlis miRma (iqiT) uhaero da usazRvro sivrceSi
moTavsebulia kosmiuri obieqtebi (mnaTobebi). dRisiT is feri, romelic cas
aqvs Sedegia imisa, rom atmosferoSi arsebuli gazebis molekulebi:
orTqli, mtveri da sxva mzis sxivebs Slian, mnaTobebis uxilaoba ki
Sedegia am movlenis gamo atmosferos Zlieri ganaTebis.
cis gumbaTi. zRvaSi an trial mindorze rom davdgeT vnaxavT, rom
Cven Tavze gvefareba cis garkveuli nawili, romlis centrSi Cven vdgevarT.
cis am nawils ewodeba cis gumbaTi.
normaluri xiluli horizonti. im moCvenebiT wreebs, romelic
Seesabameba cis gumbaTisa da dedamiwis SeerTebas ewodeba normaluri
xiluli horizonti. (nax. 1)
HH aris normaluri xiluli (moCvenebiTi) horizonti, romlis
radiusia CH=d.
C wertilSi gatarebul H H' ' Tarazul sibrtyes ewodeba WeSmariti
anu maTematikuri horizontis sibrtye. xolo TviT H H' ' xazs ewodeba
adgilis WeSmariti horizonti. xilul horizontsa da adgilis horizonts
Soris Sedgenil α kuTxes ewodeba dadablebis kuTxe.
cnobili α kuTxiT da h-iT SegviZlia gamovTvaloT dedamiwis R
radiusis miaxloebiTi odenoba formuliT:
nax. 1
( ) RhR =+ αcos (1.2.1)
RhR =+ αα coscos
αα cos)cos1( hR =−
αα
cos1cos
−=
hR (1.2.2)
H ' H '
h
O
R
Hd
α dadablebis
kuTxe
C
H
α
normaluli xiluli horizonti
adgilis WeSmariti horizonti
1111
am formuliT gamoTvlili dedamiwis radiusi 6400≈R km normaluri
xiluli horizontis d radiusi ki gamoiTvleba piTagoris TeoremiT:
2 2 2( ) 2 2 113d R h R Rh h Rh h= + + − = + = + = +
h simaRle gamoisaxeba kilometrebSi. aranormalur xilul
horizonts miviRebT mTebs Soris dgomisas. amis Sedegad normaluri
xiluli horizontis HH xazi gaivlis WeSmariti horizontis, rogorc
qvemoT, ise zemoT.
cis sfero. vinaidan varskvlavebi uaRresad Sors arian, adamianis
Tvals eCveneba, rom isini caze arian miWedili, misgan erTi da igive
manZilze. am moCvenebiT sferos, cis sfero ewodeba. cis sfero saWiroa
varskvlavebs Soris urTierT
mdebareobis gasarkvevad.
im SemTvvevaSi, rodesac cis
sferos centrad miviCnevT dedami-
waze Cveni dgomis A wertils, mas
ewodeba topocentruli cis sfero.
(nax. 2)
im SemTxvevaSi, rodesac cis
sferos centrad miviCnevT deda-
miwis C centrs, mas geocentruli
cis sfero hqvia. xolo rodesac
cis sferos centrad miviCnevT mzis
B centrs, cis sferos ewodeba
heliocentruli cis sfero.
b) heliocentruli da geocentruli
moZRvrebis arsi
sagangebo dakvirvebis gareSe SeuZlebelia dedamiwis moZraobis
dadgena, amitom adamians egona, rom uZravi dedamiwis garSemo moZraoben
amJamad cnobili, mzis sistemaSi Semavali obieqtebi (planetebi).
Cvens welTaRricxvamde II saukuneSi aleqsandrielma mecnierma
ptolomem zemoTxsenebuli midgomiT Camoayaliba e. w. geocentruli sistema.
nax. 2
1122
XVI saukuneSi kopernikma uaryo geocentruli sistema da
daamtkica piriqiT movlena, e.i. dedamiwa ki araa simravlis centri, aramed
mze, romlis garSemo sruliad normalurad, elifsuri gziT, moZraoben mzis
sistemis obieqtebi, rasac heliocentruli sistema ewodeba.
g) samyaroSi orientireba. samyaros polusebi da RerZebi.
saSuadReo xazi. zeniti da nadiri
imisaTvis, rom dedamiwaze da samyaroSi gavarkvioT wertilis
mdebareoba, saWiroa orientirebis gamoyeneba.
orientirebisaTvis iyeneben or xerxs: RamiT polarul varskvlavs
da dRisiT mzes. amisaTvis saWiroa gvaxsovdes mzis amosvlis mxare
(aRmosavleTi) RamiT. davdgebiT saxiT am mxarisaken, SevbrundebiT marcxniv
da davinaxavT didi daTvis Tanavarskvlavedis I mdebareobas
, , , , , ,α β γ σ ε ξ η varskvlavebs.
nax. 3
βα mimarTulebiT TvaliT gadavzomavT 5 aseT manZils, sadac
SevamCnevT patara daTvis α varskvlavs, es varskvlavi zustad samyaros
1133
nax. 4
Crdilo polusze ar mdebareobs. misgan moSorebulia '63 -iT. am varskvlavs
uwodes polaruli varskvlavi.
6 saaTis Semdeg didi daTvis Tanavarskvlavedi miiRebs II
mdgomareobas. kidev 6 saaTis Semdeg III mdgomareobs da a. S. saxiT α
varskvlavisaken iqneba CrdiloeTi, Cvens ukan iqneba samxreTi, marjvniv
aRmosavleTi da marcxniv dasavleTi.
dRisiT orientirebisaTvis dekretuli droiT 13 saaTze davdgebiT
saxiT mzisaken am dros mzes uWiravs umaRlesi adgili samyaroSi, Cvens win
iqneba samxreTi, ukan iqneba CrdiloeTi, marcxniv aRmosavleTi da marjvniv
dasavleTi.
1.3. koordinatebi
koordinatebs uwodeben saxeldebul fardobiT ricxvebs, romelTa
saSualebiT isazRvreba wertilTa mdebareobebi. rom vicodeT dedamiwis,
romelime wertilis mdebareoba saWiroa vicodeT am wertilis ciuri
koordinatebi.
geografiuli koordinatebi.
vTqvaT, gvinda vicodeT M
wertilis koordinatebi. davuS-
vaT Sveuli, romelic dedamiwis
centrSi ar gaivlis misi
simkvrivis sxavadasxvaobis gamo,
magram mcire SecdomiT Cven
davuSvaT, rom is gadis centrSi.
gavataroT M0 wertilis
meridianis sibrtye, romelic
ekvatoris sibrtyes gakveTs,
miviRebT ϕ brtyel kuTxes,
romelsac M0 wertilis ganedi
ewodeba. e.i. mas qmnis M0-
wertilis Sveuli ekvatoris
1144
sibrtyesTan. igi icvleba οο 900 − -mde CrdiloeTiT + niSniT, samxreTiT –
niSniT da izomeba Sesabamisi F0M0 rkaliT.
ϕ SeiZleba Seicvalos z-iT. am kuTxes ewodeba M0-wertilis
zenituri kuTxe anu polusis zenituri manZili. igi aiTvleba rkaliT ο0 -dan
ο180 -mde + niSniT e.i.
ο90=+ zϕ (1.3.1)
ϕ -gvaZlevs saSualebas gavigoT M0-wertilis paraleli. Cven ki
gvsurs M0 wertilis mdebareobis gansazRvra, e.i. saWiroa meore koordinati.
saerTaSoriso SeTanxmebiT II koordinatis gansazRvrisaTvis sawyisad
miiRes – F '0 -i (wertili miRebuli grinviCis meridianisa da ekvatoris
gadakveTiT). orwaxnagovan λ kuTxes, romelsac qmnis mocemuli wertilis
meridianis sibrtye grinviCis meridianis sibrtyesTan, ewodeba grZedi da
izomeba ο0 -dan aRmosavleTiT ο180 -mde da aris dadebiTi, ο0 -dan
dasavleTisaken ο180 -mde aris uaryofiTi. III koordinati γγ HGHMM +==0
es aris geodeziuri simaRle, romelic aris anomaliisa da normaluri
simaRlis jami.
maSasadame, ganedi ewodeba brtyel kuTxes, romelsac mocemuli
Sveuli qmnis ekvatoris sibrtyesTan. xolo grZedi ewodeba orwaxnagovan
kuTxes, romelsac mocemuli wertilis meridianis sibrtye qmnis grinviCis G
meridianis sibrtyesTan. igi izomeba orwaxnagovani kuTxis Sesabamisi F F0 0'
rkaliT. orive koordinats ewodeba geografiuli koordinatebi.
geografiuli grZedi da ganedi ϕ da λ ganisazRvreba
astronomiulad, xolo geodeziuri grZedi (B) da ganedi (L) ki gamoiTvleba.
geografiuli koordinatebiT ganisazRvreba adgilze wertilebis
mdebareoba. am koordinatebis gansazRvrisaTvis saWiroa ciuri
koordinatebis gansazRvra.
1155
geocentruli cis sfero
ciuri koordinatebi saWiroa dedamiwis fizikur zedapirze
koordinatebisa da mimarTulebebis anu azimutebis gansazRvrisaTvis.
amisaTvis vsargeblobT geocentruli cis sferoTi. cis sferoze Cven
varkvevT rkalur manZilebs. magaliTad ara αβ -swors, aramed maT gegmilebs
Soris rkals ''βα kuTxur ganzomilebebSi . (nax. 5)
ciuri koordinatebia:
1) horizontuli
2) ekvatoruli I – saxis
3) ekvatoruli II – saxis
4) ekliptikuri
am koordinatebis saxelwodeba
Seesabameba gamosavali sibrtyeebis an xazebis saxelwodebas.
nax. 5
1166
II Tavi
dedamiwis fizikuri elementebi
mz ida n m e s a m e pla n et a d ed a m iw a warmoadgens, bunebrivi,
socialuri da ekonomikuri elementebis erTobliobas.
dedamiwis bunebrivi elementebia rac arsebobs adamianis nebis gareSe,
SeiZleba maTze dakvirvebebis, gazomvebis warmoeba da zemoqmedeba
kacobriobis saWiroebisaTvis.
d ed a m iw i s bu n e br i v ele m e nt e bs warmoidgenen geosferoebis
saxiT rogoricaa: 1) dedamiwis atmosfero
2) hidrosfero
3) myari tani
2.1. a) dedamiwis atmosfero
dedamiwis atmosfero warmoadgens mTlian garss, romelic Seqmnilia
gazebis meqanikuri gaerTianebis Sedegad, mas xSirad haers uwodeben
(gavixsenoT cis ganmarteba). imis gamo, rom haeri (dedamiwis atmosfero)
SeWrilia niadagSi, wyalSi, igi dedamiwasTan erTad moZraobs, amitom
vuwodeT mas dedamiwis atmosfero (e.i. dedamiwas Tan axlavs atmosfero).
atmosferos zemo sazRvari dadgenili araa zustad: Zveli monacemebiT igi
1300≈ km-ia. mag. zRvis doneze atmosferos simkvrivea 1300 gr/m3, 10 km-ze
zeviT igi 400 gr/m3, kidev zeviT 80 km-ze – 90 gr/m3.
Tanamedrove cnobebiT, dedamiwis atmosferos sisqe 1300 km-ze metia.
adamians eCveneba, rom TiTqos atmosferos wona ara aqvs, sinamdvileSi
dedamiwis atmosfero iwonis 15105 ⋅ t=5000000 miliard tonas. atmosferos
mTliani garsi, rom foladis simkvrivemde SevkumSoT dedamiwas
Semoertymeboda 12.5 m sisqis javSani (yovel Cvengans 0.1≈ t haeri awveba).
atmosfero aris adamianis arsebobis erTerTi ZiriTadi saSualeba.
misi Semadgeneli Jangbadi, azoti, naxSirbadi da sxva brunavs organuli
bunebidan araorganul bunebaSi.
haeri adamians icavs mzis mavne gavlenisagan (mag. mavne gavlenaa
ultraiisferi gamosxiveba), amave dros wvas xels uwyobs.
1177
atmosferos yofen Sreebis mixedviT, romelsac safuZvlad udeben
kriteriumebs Semadgenlobis an temperaturis mixedviT. magaliTad,
Semadgenlobis mixedviT:
ozonosfero, ionosfero
an kidev temperaturis mixedviT:
1. troposfero – tropopauziT
2. stratosfero – stratopauziT
3. mezosfero – mezopauziT
4. Termosfero – TermopauziT
5. egzosfero – egzopauziT
saerTod pauzis sisqe (Weri) aris 200 m-dan 2 km-mde.
troposfero sxvadasxva sisqisaa dedamiwis sartyelebis Sesabamisad,
magaliTad, ekvatoris sartyelTan misi sisqea 16-18 km. saSualo (Cveni)
sartyelis Sesabamisad 12-16 km. da polusebis Sesabamisad 8-12 km. Cveni mTebi
moTavsebulia troposferoSi. troposfero iWers mzis gamosxivebis 15%. aq
aris mTeli atmosferos 75%. TviT dedamiwa iWers mzis gamosxivebis 50%-
mde. maSasadame: pirvel rigSi mzisgan dedamiwa Tbeba da Semdeg
dedamiwidan Tbeba troposfero (inversiis uwesrigobis Sedegad xandaxan
dedamiwa civia, haeri Tbilia).
tropopauzidan _ zemoT 60 km-mde aris stratosfero. aq temperatura
(35 km-mde) kidev iklebs da minus ο60 -mde dadis. 35 km-dan zemoT 60 km-mde
temperatura + ο20 -mde gaizrdeba, radgan am nawilSi bevri ozonia (ozoni
bevr siTbur energias STanTqavs).
stratopauzidan vertikalurad zeviT 80 km-mde Sres ewodeba
mezosfero. aq ozoni TiTqmis araa da - ο80 , - ο90 -mde dadis temperatura.
mezopauzidan 800 km-mde Sres ewodeba Termosfero. aq temperatura
aRwevs + ο500 .
Termopauzidan 1300-1500 km-mde Sres ewodeba egzosfero, sadac
temperatura aRwevs + ο1500 -mde. ο1500 -ze Termuli energia naklebia, vidre
CvenTan, radgan iq haeri gaiSviaTebulia (molekulebma unda gaiaros 20-30
km, rom erTmaneTs daejaxos).
saerTod, atmosfero astronomiuli dakvirvebebisaTvis metad
xelSemSlelia, amitomaa, rom Tanamedrove mecniereba ibrZvis mTvaris
dapyrobisaTvis, rom masze iqnes gaSenebuli xelovnuri, kosmiuri obieqtebi.
1188
b) hidrosfero
igi warmoadgens dedamiwis wyvetil garss, romelic Sedgeba msoflio
okeaneebisagan (Tavisi zRvebiT), tbebiT da mdinareebiT.
msoflio okeaneebs uWiravs mTeli teritoriis 70.2%, xmeleTs ki 29.8%.
msoflio okeaneebs warmoadgenen: napiridan 200 m-mde okeanuri
nariyebi, Selfi 200 m-dan 24002000 −≈ m-mde okeanuri ferdobebis, 6000m mde
okeanuri sadinarisa (kalapotisa) da 11 km-mde okeanuri Rrmulebis saxiT.
g) dedamiwis myari tani
warmogvidgeba geosferoebis saxiT, rogoricaa:
1. qerqi – liTosfero 7-dan 70 km-mde
2. zemo mantia – 800 km-mde (e.i. misi sisqea 730 km)
3. mantia anu saSualo Sre 800 km-dan 3500 km-mde
4. guli – 3500 km-dan 6400 km-mde
faunisa da floris anu sicocxlis arsebobis TvalsazrisiT
troposferos, hidrosferos da myari tanis 3-4 km-s erTad uwodeben
biosferos (e.i. Sedis samive), rac niSnavs imas, rom iq arsebobs sicocxle.
d) dedamiwis socialuri elementi
kacobrioba – ganlagebuli dedamiwaze sxvadasxva erovnebebis,
mosaxleobis saxiT.
e) dedamiwis ekonomikuri elementi
yovelive, rac Seqmnilia kacobriobis mier. magaliTad, mrewvelobis,
soflis meurneobis, kavSirgabmulobis, janmrTelobis da sxva obieqtebi
dedamiwis ekonomikur elementebs warmoadgenen.
dedamiwis zemoxsenebul bunebriv, socialur da ekonomikur
elementebs uwodeben dedamiwis fizikur elementebs. mokled dedamiwis
elementebs.
warmoebis Seswavlisa da analizis TvalsazrisiT dedamiwis
elementebs kidev uwodeben qveynis sawarmoo Zalebs.
1199
qveynis sawarmoo Zalebi anu cocxali samuSao Zala da sawarmoo
saSualebebi.
cocxali samuSao Zala – dedamiwis socialuri elementia.
sawarmoo saSualebebia: teqnika, Senobebi (dedamiwis ekonomikuri
elementebi) da bunebrivi simdidreebi.
v) dedamiwis fizikuri zedapiri
xiluli zedapiri. dedamiwis namdvili saxe. wiaRi. adgili.
wvliladebi.
dedamiwis fizikuri zedapiri ewodeba misi liTosferos zedapirs
mTlianad, romelic aerTianebs xilul da uxilav nawils. xiluli nawilia
xmeleTis zedapiri, romelsac sazRvravs atmosfero da uxilavi nawilia
okeaneTa fskeri.
dedamiwis xiluli zedapiri warmoadgens xmeleTisa da okeaneTa
zedapirebis erTobliobas.
dedamiwis namdvili saxe aris xiluli zedapiri mTlianad.
wiaRs vuwodebT myari tanis Siga uxilav nawils, romelic ar sCans.
adgils vuwodebT Sesaswavlad gamiznul sivrcis nawils, romelic
moicavs dedamiwis elementebs.
wvliladebi ewodeba, adgilis cnebis Sesabamisad (im sagnebs)
dedamiwis im elementebs, romlis Seswavla da agegmvaa SesaZlebeli.
z) dedamiwis topografiuli elementebi
adgilis ukeT Seswavlis mizniT, wvliladebis cnebis Sesabamisad,
dedamiwis elementebs anu qveynis sawarmoo Zalebs geodeziaSi vuwodebT
dedamiwis topografiul elementebs, romlebsac warmoidgenen:
1. topografiuli zedapiris (bunebrivi elementi)
2. samTo qanebi, gruntebi da niadagebis (bunebrivi elementi)
3. hidrografiis (bunebrivi, xelovnuri elementi)
4. mcenareuloba (bunebrivi, xelovnuri, ekon. elementebi)
5. gzebi (xelovnuri anu ekonomikuri elementebi)
6. mosaxleobis saxiT.
A1
A2
A3
A B C
2200
to pografiuli z ed a p ir i . topografiuli zedapiri ewodeba
dedamiwis fizikuri zedapiris nawils, romelsac Sveuli xazebi mxolod
erT wertilSi kveTen. topografiuli zedapiris funqcia
),( yxfZ = xasiaTdeba calsaxobiT, sasrulobiT, uwyvetobiTa da
mdovrulobiT. (nax. 6)
dedamiwis fizikuri zedapiris calkeul uswormasworobebs, romelTac
aqvT garkveuli garegani saxe, uwodeben r eli ef is a nu
topografiuli z eda p ir i s form e b s . reliefis ZiriTadi
form e b i a (nax. 7):
nax. 7
gora – topografiuli zedapiris amoburcul-amaRlebuli adgili.
misi elementebia: Ziri, gverdebi, wvero. gora, romlis wveroc wvetanaa
2211
ewodeba wvetiani gora. im goras, romelic zvins waagavs uwodeben
mrgvalTavian goras. waWrilTavian goras ewodeba zegani. mcire goras
ewodeba borcvi, kidev ufro mcire xelovnur goras yorRani.
goras Sebrunebuli aris t afobi , dedamiwis CaRrmavebuli adgili.
misi elementebia: fskeri, gverdebi da piri. mcire tafobs ewodeba ormo.
mTis ferdobze amaRlebul wagrZelebul adgils ewodeba q edi . misi
elementebia: wyalgamyofi xazi anu dadebiTi invariantuli xazi da
ferdobebi.
dadebiT invariantul xazze rom davdgeT, Cvengan erTi mimarTulebiT
maRldeba, sami mimarTulebiT dabldeba.
R ele aris (qedis Sebrunebuli adgili) mTis ferdobis CaRrmavebuli
wagrZelebuli adgili. misi elementebia: wyalmaerTi anu uaryofiTi
invariantuli xazi da ferdebi. uaryofiTi invariantuli xazi Sebrunebulia
dadebiTi invariantuli xazisa. igi erTi mimarTulebiT dabldeba, sami
mimarTulebiT ki maRldeba.
mcire ferdebCamoglejil Reles – xra m s , x e v s , Rra nt e s
uwodeben.
mTebs Soris moTavsebul did sivrces, sadac moTavsebulia
mosaxleoba, miedineba mdinare, aris naTesebi ewodeba x eob a .
xeobis im adgils, sadac mTebi TiTqos Seirwya (erTdeba) ewodeba
klde - k ar i – gasasvleli.
mTis ferdobebze wavakebul adgils ewodeba s af e xura (t er a s i ) .
mTis ferdobze im moedans, sadac erTdeba ori qedi da iwyeba ori
Rele ewodeba u n a g ir a (e.i. wyvili invariantuli xazebi erTdeba).
mTebis xazovan erTobliobas uwodeben mT agre x ils , mis umaRles
SemaerTebel xazs ewodeba x erx e m ali .
iq sadac xerxemali Cazneqilia ewodeba uReltex ili ( g ad a -
s a s vleli ) .
mTebis wyalgamyof xazebs Soris moTavsebul sivrces, romlis
naleqebiTac maragdeba ZiriTadi mdinareebi ewodeba auzi .
III Tavi
dedamiwis namdvili saxisa da odenobis Seswavlis
2222
(mcdeloba) cda.
3.1. geodeziis sagani
praqtikis moTxovnis Sesabamisad Tavisi arsebobis periodSi
kacobriobam Sehqmna sxvadasxva samecniero dargebi. xelovnebis dargebis
Seqmnac adamianis sulieri, bunebrivi moTxovnilebis Sedegia.
amave dros, saWiro Seiqmna qveynierebaze zogadi warmodgenis
(msoflmxedvelobis) Seqmna, romelic meTodologiur safuZvlad daedeboda
mecnierebisa da xelovnebis sxvadasxva dargebs, aseT mecnierebas ewodeba
filosofia.
maSasadame, mecniereba, erTis, mxriv Seiswavlis samyaros, rogorc erT
mTlians, filosofiis (mecniereba bunebis, sazogadoebis da azrovnebis
ganviTarebis uzogadesi kanonebis Sesaxeb) saxiT, meores mxriv, sxvadasxva
TvalsazrisiT, samecniero dargebis saSualebiT, rogoricaa: teqnikuri,
sabunebismetyvelo, humanitaruli da sxva.
geodezia aris sabunebismetyvelo da sainJinro samecniero dargi,
sadac dedamiwis bunebrivi elementebis mravalnairi gazomvebis, ganazomTa
maTematikuri damuSavebis da gamonaTvalTa grafikulad gamoxazvis
saSualebiT (rukebi, gegmebi, profilebi) Seiswavleba:
dedamiwis namdvili saxe (forma) da odenoba (zomebi), droTa
viTarebaSi maT saxecvalebadobasTan dakavSirebiT; misi bunebrivi,
socialuri, ekonomikuri da topografiuli elementebis sivrcobrivi
ganlagebebi da zomebi; geometriulad sworad agebis TvalsazrisiT,
swyvets saxalxo meurneobisa da samxedro strategiis iseT amocanebs,
romlebic dakavSirebulni arian sainJinro nagebobaTa da sasoflo
sameurneo daniSnulebis obieqtTa kvleva-Zieba-trasirebasTan,
proeqtirebasTan, mSeneblobasTan, eqsploataciasTan da samxedro saqmesTan.
(aseTia geodeziis Tanamedrove ganmarteba).
ratom aris geodezia sabunebismetyvelo dargi? imdenad, ramdenadac
geodeziaSi Seiswavleba dedamiwis namdvili saxe da misi odenoba, amdenad
is sabunebismetyvelo dargia.
rogorc sainJinro samecniero dargSi, geodeziaSi ZiriTadad wydeba 5
amocana:
1. topograful elementebze dakvirvebebi da gazomvebi.
2233
2. ganazomTa maTematikuri damuSaveba da maTi grafikulad gamoxazva
rukebis, gegmebis, profilebis saxiT, rasac mokled ewodeba
ganazomTa kameraluri damuSaveba.
3. arsebuli rukebisa da gegmebis gamoyeneba sxvadasxva sainJinro
amocanebis gadasawyvetad.
4. proeqtis monacemTa gegmebidan adgilze gadatana (dakvalva);
Senebis procesSi zedamxedveloba da uSualo monawileobis
miReba yvela konstruqtuli elementebis geometrulad sworad
Senebis TvalsazrisiT da SesrulebiTi naxazebis sworad Sedgena.
5. nagebobaTa eqsploataciis procesSi gegmebis koreqtireba,
nagebobebis deformaciebze da dacurebebze dakvirvebebis warmoeba
da sxva.
rac Seexeba samxedro saqmes, amboben ruka armiis Tvaliao.
geodeziis sagnis zemoT gansazRvrebis Sesabamisad arsebobs cikli
geodeziuri samecniero dargebisa, rogoricaa: umaRlesi geodezia,
kartografia, geodeziuri astronomia, geodeziuri gravimetria, aerogeodezia,
topografia, sainJinro geodezia, samarkSeidero saqme, samTo geometria,
topografiuli da samarkSeidero xazva kaligrafiiTurT, instrumen-
tTmcodneoba, ganazomTa maTematikuri damuSaveba, romelSic Sedis
ganazomTa Secdomebis Teoria, gamoTvlebis teqnika, umcires kvadratTa
meTodi, ganazomTa albaTobis Teoria da maTematikuri statistika.
3.2. dedamiwis namdvili saxisa da odenobis
dadgenis problemis arsi
rogorc cnobilia, dedamiwis namdvili saxea misi xiluli zedapiri
mTlianad. es zedapiri urTulesia, misi elementebis maTematikuri
formulebiT gamoTvla SeuZlebelia, ris gamoc mivmarTavT generalizacias
(ganzogadoebas). magaliTad Cven vzomavT AB umokless manZils da ara
namdvil manZils (nax. 8).
2244
L0
L0
nax. 8
A B
dedamiwis namdvili saxis damaxasiaTebeli wertilebis sam-sami
koordinatis gansazRvra mas-
ze gazomvebis saSualebiT
SeiZleboda, magram Zalian
garTuldeboda gamoTvliTi
samuSaoebi, amave dros samu-
Saoebi damokidebulia TviT
zedapiris formebis gamoyene-
basTan, rac ucnobia. maSasa-
dame ucnobis ucnobiT gansaz-
Rvris surviliT mizans ver
mivaRwevT. aRniSnulis gamo saWiro gaxda:
1. dedamiwis namdvili saxidan gamoiyos misi udidesi nawili,
romelsac dedamiwis saerTo saxe vuwodoT.
2. saerTo saxis zedapiri unda gamoyenebuli iqnes fizikur
zedapirze ganazomebis reducirebisaTvis (sagangebod dagegmilebisaTvis) da
maTematikuri damuSavebisaTvis, rac mogvcems saSualebas ganvsazRvroT
yoveli wertilisaTvis or-ori koordinati (ganedi da grZedi)
3. es zedapiri unda gamoyenebuli iqnas rogorc Sesadari (sawyisi,
gamosavali) mesame koordinatis gansazRvrisaTvis.
a) dedamiwis saerTo saxe
ZvelTagan dedamiwis saerTo saxed Tvlidnen da Tvlian im idealur
zedapirs, romelsac miviRebT wynar mdgomareobaSi myofi (warmodgeniT)
okeaneebis saSualo donis gangrZobiT xmeleTis qveS urTierT Serwymamde
(SeerTebamde) ise, rom yovel wertilSi Sveuli xazebi warmoadgendnen
normals (e.i. Sveuli xazebi iyos marTobuli dedamiwis zedapiris).
dedamiwis saerTo saxis, rogorc sferos I-miaxloebiT gansazRvrisas
piTagoras skola Cv. w. aR. VI saukuneSi, aristotele Cv. w. aR. IV saukuneSi,
arqimede Cv. w. aR. III saukuneSi Tvlidnen, rom dedamiwis saerTo saxe sfero
iyo, ris dasadastureblad iyenebdnen Semdeg movlenebs:
2255
nax. 9
1. mTvaris dabnelebis dros dedamiwis Crdili mTvareze wrea.
2. normaluri xiluli horizonti wriulia
3. amerikis aRmoCena dedamiwis sferulobis rwmenis Sedegia
4. trial mindorze svlisas, nagebobebTan an tyesTan miaxlovebisas
jer wveri Cndeba, daSorebisas pirvelad Ziri uCinardeba.
b) dedamiwis, rogorc sferos, radiusis
gansazRvris pirveli cda
Cv. w. aR. 230 wels aleqsandrielma mecnierma eratosTenem pirvelma
gazoma dedamiwis radiusi Semdegi midgomiT.
man warmoidgina, rom siena (asuani) da aleqsandria (sadac
cxovrobda) erT meridianzea. sinamdvileSi ki isini dacilebulia ο3 .
ϕs -s sienas astronomiuli ganedi, xolo
ϕ a-s aleqsandrias astronomiuli ganedi ewodeba (nax. 9)
misi mizani iyo gaezoma manZili S da οδ kuTxe. maSin advilad
gansazRvravda radiuss.
2266
δ ο R
S
nax. 10
manZili S gansazRvra aqlemebis qaravnebis saSualo siCqariTa da
droiT. gamovida 800≈ km. οδ -kuTxis gasagebad yovelwliurad 22-ivniss
(zafxulis mzebudobis dRes) Rrma Widan mze Canda, anu mze zenitSi iyo e.i.
Z siena=0.
maSasadame h s=ο90 ,
radgan Z s+hs=ο90 (3.2.1)
imave dRes aleqsandriaSi dakvirvebis dros Crdili mzeze
mimarTulebidan gadaixara sruli wrexazis 501
nawiliT, e.i. misi zenituri
manZili
Zaleq = =360 50 7 2ο ο: ,
maSin haleq -= =90 7 2 82 8ο ο ο, ,
e.i. S rkalis Sesabamisi kuTxe δ kuTxe iqneba
δ ϕ ϕο ο ο ο ο ο= − = − =aleq siena aleq sienaZ Z 7 2, (3.2.2)
anu
δ ϕ ϕο ο ο ο ο ο= − = − =aleq siena aleq sienah h 7 2, (3.2.3)
rogorc vxedavT S-i gansazRvrulia geodeziurad,
xolo Z0-astronomiulad. davwerT
SR
=δρ
ο
ο aqedan R S= ⋅ρδ
ο
ο , ρπ
=180ο
(3.2.4)
ρο-radianis zomaa ", "
157 3 3438' 206265sin1
= = = =o . imis
gamo, rom es meTodi dResac gamoyenebulia, eratosTene
iTvleba geodeziis mamamTavrad.
3.3. cneba gradusuli gazomvebis Sesaxeb
gradusuli gazomvebis ZiriTadi principia: erTi gradusis Sesabamisi
rkali, ise Seesabameba nebismier rkals, rogorc erTi gradusis Sesabamisi
centraluri kuTxe am nebismieri rkalis centralur kuTxes.
2277
ο
οο
δ11 =
SS
, SS ⋅= ο
ο
ο
δ1
1.
udidesi gradusuli gazomvebis Sesrulebis safuZvelze
rwmundebodnen imaSi, rom ο1 -rkalebis sigrZe dedamiwis polusebTan ufro
grZeli gamodioda vidre ekvatorTan. droTa viTarebaSi darwmundnen, rom
dedamiwa sferoidulia. sabolood niutonis msoflio mizidulobis
kanonis safuZvelze dadasturda, rom dedamiwis saerTo saxe aris, sferoidi,
(brunvis elifsoidi).
a) triangulaciis gamoyenebis pirveli cdebi
dedamiwis namdvili saxis metismeti sirTule mudam xelis SemSleli
iyo xazebis gazomvebis dros, rac sagrZnoblad gaumjobesda manZilebis
arapirdapiri xerxiT gansazRvris gamogonebis
Sedegad. es xerxi pirvelad gamoiyena
danielma astronomma tixo-bragem 1578 wels,
magram mis fuZemdeblad sTvlian holandiel
mecniers sneliuss (1580-1626), romelmac or
wertils Soris gansasazRvreli manZilis
Sesabamis sivrceze aago samkuTxedebis qseli,
romelSic gaizoma sami mcire zomis bazisi, anu
fuZe (aucilebeli iyo mxolod erTi bazisi),
da yvela kuTxe am samkuTxedebisa. sinusebis
TeoremiT pirveli gverdisa (bazisis) da
gazomili kuTxeebis gamoyenebiT gansazRvra qselis samkuTxedebis yvela
danarCeni gverdis sigrZe, ramac saSualeba misca am wertilebs Soris
manZilis gansazRvrisa. xazebis sigrZeebis gansazRvrisaTvis saWiro
samkuTxedTa qsels da am qselis Sedgenis meTodsac ewodeba
triangulacia. am meTodis pirvelyofili saxe ganazomTa, gamonaTvalTa da,
saerTod, samkuTxedTa qselis saTanado analizis gareSe sruldeba.
magaliTad (nax. 11), meridianis urTierT uxilavi boloebisa da didi
dabrkolebis momcveli AG rkalis sigrZis gansazRvrisaTvis misi Sesabamisi
G
EE
FF
DD
CCBB
AAnax. 11
2288
sivrcis winaswari daTvalierebis (rekognoscirebis) Sedegad SearCeven
uSualod advilad gasazom mokle AB xazs, romelsac bazisi (fuZe) vuwodeT,
da agreTve daniSnaven samkuTxedTa wveroebs im varaudiT, rom imzirebodes
samkuTxedis yoveli wverodan, rogorc danarCeni ori wvero, aseve masTan
dakavSirebuli Semdegi samkuTxedebis wveroebi. uSualod zomaven AB baziss
da yvela samkuTxedis Siga kuTxeebs da sinusebis Teoremis gamoyenebiT
sazRvraven AC, BC, BD, CD, DE, CE, DF, EF, EG, FG gverdebs; Semdeg cnobili
(gansazRvruli) DE gverdisa da Sesabamisi kuTxeebiT sazRvraven AE, AD, DG
gverdebs; am gverdebisa da saTanado kuTxeebis saSualebiT isazRvreba AG
rkalis sigrZe orjer, ris Sesabamisad moiTxoveba daculi iyos Semdegi
toloba:
AG AE AEGEGA
AD ADGAGD
= =sin ∃
sin ∃sin ∃
sin ∃ (3.3.1)
cxadia, sakiTxis ase gadawyveta gamoTvlis kontrolisa da saimedo
Sedegis saSualebas iZleva. am xerxis Sedegad amaRlda xazovani gazomvebis
done; kuTxuri gazomvebis done ki SedarebiT maRali iyo. triangulaciis
meTodi amJamad farTod gamoiyeneba, rogorc dedamiwis figurisa da
zomebis dadgenisaTvis saWiro gradusuli gazomvebis, ise saxelmwifo
kartografiuli da sagangebo agegmvebisaTvis, agreTve dakvalvebisaTvis
dasayrden qselTa Seqmnis saqmeSi.
triangulaciis meTodi warmatebiT gamoiyena pikarma (1620-1682),
romelic 1669-1670 wlebSi safrangeTis mecnierebaTa akademiis davalebiT
asrulebda parizsa da amiens Soris rkalis gazomvebs. am gazomvebis
Sedegad 1 2355ο ' " rkalis sigrZe miRebul iqna 153689 m, saidanac 1ο rkals
Seesabameba 111,212 km, xolo R=6371692 m. Tanamedrove gazomvebiT igive
rkalis 1ο sigrZea 111,221 km. vinaidan gazomvebis teqnika maSin dabali iyo,
aseTi TanmTxveva SemTxveviTad iTvleba, magram mainc aRsaniSnavia, rom
gazomvebi maSindeli droisaTvis Sesrulebuli iyo maRal doneze. pikari
Tarazuli kuTxeebis gazomvis dros iyenebda ZafTa badiT aRWurvil Wogrs,
xolo ganedebis gazomvisaTvis – zenitur seqtorebs.
cnobilia, rom niutonma aTeuli wlebis ganmavlobaSi Tavi Seikava mis
mier aRmoCenili msoflio mizidulobis kanonis gamoqveynebisagan (1687 w.),
vidre ar Seamowma mTvaremde am kanonis gavrcelebis siswore, risTvisac
pikaris mier miRebuli dedamiwis radiusis zusti mniSvneloba gamoiyena.
2299
pikaris mier Sesrulebuli samuSaoebiT SeiZleba damTavrebulad
CaiTvalos dedamiwis saerTo saxis zomebis dadgenisaTvis SedarebiT
mecnierul doneze dayenebuli saWiro gazomviTi samuSaoebis pirveli
periodi, romelic moicavs daaxloebiT 2000 wels da romlis dros
dedamiwis saerTo saxe warmodgenili iyo mxolod rogorc sfero Sveuli
xazebis wrfeebad warmodgenis safuZvelze, magram aqve saWirod migvaCnia
aRvniSnoT, rom amave periodSi safuZveli eyreba dedamiwis figuris Teoriis
dadgenasTan dakavSirebul gravimetriul (fizikur) gazomvebs. niutonis
Tanamedrove frangma astronomma riSem (1640-1696) 1672 wels ekvatoris
axlos, samxreT amerikaSi mdebare kanionSi, roca is awarmoebda marsis
paralaqsis gansazRvrebs, SeniSna, rom parizSi Semowmebuli da dayenebuli
qanqariani saaTi dRe-RameSi ukan rCeboda 2,5 minutiT da swori msvlelobis
aRsadgenad saWiro gaxda qanqaris sigrZe daemoklebinaT 3 mm. analogiur
movlenebs amCnevdnen sxva mogzauri mecnierebic, magram vercerTma ver
axsna am movlenis mizezi. mxolod niutonma (1643-1727) SesZlo aexsna
SeniSnuli movlena polusebidan ekvatorisaken simZimis Zalis Semcirebis
mizeziT.
b) dedamiwis saerTo saxis dadgena
gazomvebis dros gamoTvlis Sedegad iRebdnen erTi da igive
meridianis sxvadasxva wertilebis grZedebis da aseve paralelis mimarT
ganedis sxvadasxva mniSvnelobebs. amas abralebdnen gazomvis SemTxveviT
Secdomebs, magram Semdeg gamoirkva rom es ase ar aris. amis mizezia
Sveulis gadaxra.
droTa viTarebaSi darwmundnen, rom dedamiwis saerTo saxe aris
sferoidisagan gansxvavebuli tani – geoidi.
didi xnis manZilze fiqrobnen, rom brunviTi elifsoidis (sferoidis)
a didi naxevarRerZisa da α SekumSulobis gansazRvris sizusteze gavlenas
axdens S-rkalis sigrZisa da ϕ ganedis an λ grZedis Secdomebi.
maSasadame, Tvlidnen, rom Sveuli xazebi wrfeebs warmoadgenen da ar
iRebdnen mxedvelobaSi imas, rom es Sveulebi sinamdvileSi gadaxrilebi
arian dedamiwis tanSi masebis sxvadasxva simkvrivis gamo. e.i. erTi da igive
meridianis an paralelis sxvadasxva wertilebis koordinatebis
3300
urTierTgansxvavebuli mniSvnelobebis mizezia zemoxsenebuli elementebis
gazomvis SemTxveviTi Secdomebi.
sabolood dadginda, rom aRniSnuli Secdomebis mizezia Sveulis
gadaxrebi, mag. Sveulis gadaxra 27", gamodis, xolo SemTxveviTi Secdomis
Sedegia "3 .
dedamiwis saerTo saxis gamomsaxveli Ziebuli zedapiri
sferoidisagan (romelic xasiaTdeba mxolod sul mTlianobiT da sul
amoburculobiT) unda gansxvavdebodes sul horizontalurobiTac.
maSasadame, Ziebuli zedapiri yovel wertilSi perpendikularuli
iqneba Sveuli xazebisa, rogorc mrudeebisa, e.i. marTobuli unda iyos ara
wrfivi xazebisa.
aseT zedapirs germaneli mecnieris listingis winadadebiT (1873w)
uwodes geoidis zedapiri, TviT tans ki geoidi.
Cven viciT, rom donebrivi zedapiri SegviZlia gavataroT usasrulo
raodenobisa, rogorc liTosferos Sig, ase mis gareT.
dedamiwa, rom uZravi iyos, ar hqondes adgili mTebs, gorebs, Rrublebs,
nivTierebaTa simkvriveebic erTgvarovani iyos, maSin misi saerTo saxe
iqneboda sfero da Sveuli xazebi centrSi gaivlidnen, magram arcerTi
zemoxsenebuli piroba Sesrulebuli ar aris. amitom Ziebuli zedapiris
forma iqneba sferosagan gansxvavebuli tani. aqve SeiZleba avRniSnoT, rom
okeaneebi warmoadgenen saSualo pireuls, mudam marTobs Tarazuli
mimarTulebis mimarT. xmeleTis amplitudaa (everesti – 9km, okeaneTa
Rrmuli – 11km)≈20km. igi dedamiwis saerTo saxesTan umniSvneloa da 1m-iani
diametris dedamiwis maketze 0.7mm gamoCndeboda.
zemoT aRniSnulis safuZvelze geoidis zedapiri unda xasiaTdebodes
sul mTlianobiT, sul amoburculobiT da sul horizontalurobiT.
rogorc aRvniSneT, donebrivi zedapirebi SeiZleba gatardes uamravi,
romlebic ZiriTadi donebrivi zedapiris paralelurebi ar iqnebian da arc
urTierTSoris paralelurebi arian, imis gamo, rom Zaluri xazebi
paralelurebi ar arian.
90ο 90ο 90ο O1
O2
O3 O4 O5 O6
O7
U1=0
U2 ≠ 0
U3=0
U4 ≠ 0 U5 ≠ 0 U6 ≠ 0
U7=0
3311
aviRoT okianuri zedapiri, romelic emTxveva sferoids, 1O wertilSi,
romlis normali elifsoidisadmi adgens ο90 kuTxes. Sveuli emTxveva
normals da gadaxra 1 0U = . 1U -s ewodeba Sveulis astronomogeodeziuri
gadaxra.
2O -wertilSi normali ar gayveba Sveuls da gadaxra 2 0U ≠
3O - 3 0U = Sveuli gayveba normals 04 ≠O 5O da 6O 0≠U Sveuli
gadaixara tanSi arsebuli wiaRis simkvrivis sxvadasxvaobis gamo, 07 =O .
ismis kiTxva, SeiZleba Tu ara gamoviyenoT geoidi fizikur zedapirze
ganazomTa reducirebisa da maTematikuri damuSavebisaTvis nax. 20-ze
21 HH ≠ . es imas niSnavs, rom simaRle damokidebulia imaze, Tu saidanaa igi
gansazRvruli. amis mizezia is, rom geoidis donebrivi zedapirebi 3O
wertilSi 0U = Sveuli gayveba normals, 4U , 5U da 6U wertilebSi 0U ≠ , 7O
wertilSi ki 0U = . araparalelurebi arian erTmaneTis mimarT, amitom A da
B wertilebis simaRleebi sxvadasxvaa.
3.4. msoflio elifsoidi. referenc-elifsoidi
nax. 13 A
A0
B
H1
B0 H2
(4)
H geoidi
Z.d.z.
(3)
3322
imis gamo, rom geoidis zedapiris gamoyeneba gegmilT zedapirad ver
xerxdeba, iZulebuli varT mivmarToT aproqsimacias, rac niSnavs gegmilT
zedapirad miviRoT maTematikuri zedapiri, romlis parametrebis gansazRvra
SegviZlia da axlosaa geoidTan. cxadia, aseT sxeulad unda miviRoT
sferoidi, brunvis elifsoidi msoflio elifsoidis saxiT, romelic unda
xasiaTdebodes:
1. misi centri unda emTxveodes dedamiwis simZimis centrs;
22.. ekvatori emTxveodes geoidis ekvators;
33.. moculoba udrides geoidis moculobas;
44.. geoididan gadaxraTa kvadratebis jami unda iyos minimaluri.
maSasadame, igi unda gamodges mTeli dedamiwis an misi udidesi
nawilebis reducirebis, maTematikuri damuSavebis da sasimaRlo safuZvlad.
radgan aseTi absoluturi msoflio elifsoidis parametrebis
dadgenisaTvis jer Sesrulebuli ar aris msoflio gradusuli gazomvebi,
anu astronomo-geodeziuri (geometriuli), fizikuri (gravitaciuli) da
sufTa astronomiuli gazomvebi, amitom msoflio (saerTo) elifsoidi jer-
jerobiT ar arsebobs.
zemoxsenebulis gamo saxelmwifo an saxelmwifoebi Tavis teri-
toriaze Sesrulebuli udidesi gradusuli gazomvebis Sedegad sazRvraven
samuSao elifsoidis parametrebs da iyeneben TavianT teritoriebze
ganazomTa reducirebis, maTematikuri damuSavebis da sasimaRlo
safuZvlad.
samuSao elifsoidebi xasiaTdebian imiT, rom maTi gadaxraTa
kvadratebis jami geoididan mocemul saxelmwifos teritoriaze
minimaluria. aseT elifsoids uwodeben referenc_elifsoids.
referenc_elifsoidebi mravalia: beselis, heifordis, klarkis,
krasovskis da sxva.
CvenSi dRemde miRebuli iyo krasovskis referenc_elifsoidi, romlis
parametrebi gamoTvlilia krasovskis xelmZRvanelobiT. daaxloebiT
gadawyvetili iqna 1500-mde ucnobiani gantolebebi, sadac gamoiyenes sabWoTa
kavSiris, indoeTis, CineTis da evropis qveynebis ganazomebi da
gravitaciuli monacemebi.
3333
nax. 14
a – didi naxevarRerZi
b – mcire naxevarRerZi
α - SekumSuloba
a
abα −
= (3.4.1)
krasovskis elifsoidis parametrebia:
6378245=a m
3.298
1=α
meoce saukunis samocian wlebSi, amerikuli da sabWoTa
TanamgzavrebiT dadginda, rom 31.298
1=α ,, romelic yvelaze axlos iyo
krasovskis monacemebTan.
dedamiwis elifsoidis erTiani sayovelTao zomebi pirvelad
damtkicebul iqna saerTaSoriso geodeziuri da geofizikuri kavSiris XVI
asambleis mier grenoblSi (safrangeTi) 1975 wels:
didi naxevarRerZi a 6378140 5= ± m;
SekumSuloba =1/298,257α ;
momdevno wlebSi zemoaRniSnuli parametrebi zustdeboda:
1983 wels – a=6378137 1± m, =1/298,256α ;
1987 wels – a=6378136 1± m, =1/298,256α ;
E1 E
PS
PN
O
b
a
3344
nax. 15
IV Tavi
geodeziuri samuSaoebis mokle mimoxilva
4.1. ZiriTadi geodeziuri samuSaoebi
dedamiwis namdvili saxisa da misi topografiuli elementebis
geometriulad Seswavlis mizniT geodeziaSi sruldeba ZiriTadi
geodeziuri samuSaoebi da agegmvebi.
ZiriTadi geodeziuri samuSaoebis Sesasruleblad Sendeba sagangebo
punqtebi da samizne niSnebi piramidebis, signalebis da sxvaTa saxiT, rasac
mokled saxelmwifo geodeziur qsels uwodeben.
maRali sizustis saxelmwifo geodeziur qselze dayrdnobiT iqmneba
dabali sizustis geodeziuri qselebi, ris safuZvelzec warmoebs agegmviTi
samuSaoebi.
ZiriTadi saxelmwifo geodeziuri qselis horizontaluri safuZvlis
Seqmna anu punqtebis horizontaluri koordinatebis gansazRvra xdeba
astronomiulad ϕ ganedisa da λ grZedis da geodeziurad B ganedisa da
L grZedis saSualebiT.
astronomiuli xerxiT koordinatebis gansazRvra moxerxebulia imis
gamo, rom adgilze punqtebis mSenebloba da manZilebis gazomvebi ar
sruldeba. xdeba mxolod mnaTobebze dakvirvebebi da Tu xilvadoba kargia,
Sesasruleblad advilia, magram Tanxlebulia SecdomebiT, rac mcire
sivrceebze misaRebi araa.
magaliTad, punqtis ϕ ganedis gansazRvraSi 20", Secdoma adgilze
gvaZlevs daaxloebiT 6m. xazovan
cdomilebas ⎟⎠⎞
⎜⎝⎛ ≈⋅ 66371110
2062652.0
,
garda amisa Sveulebis gadaxra
normalidan daaxloebiT "5 -iT,
igive gamoTvliT gvaZlevs 150m
Secdomas. amitom astronomiul
meTodTan erTad iyeneben gra-
vimetriul meTodsac, rac Secdo-
mas igive pirobebSi amcirebs "1 -mde. "1 -s ki Seesabameba 30m. xazovani
A
B 1 2
6
7 8
12
3
4 5
9
10 11
C E
D F
3355
Secdoma. mcire teritoriebze es Secdomac Zalian sagrZnobia da amitom am
xerxs ar mimarTaven da sargebloben geodeziuri meTodiT.
geodeziuri meTodiT koordinatebis gansazRvrisas sargebloben: 1.
triangulaciis 2. trilateraciis da 3. poligonometriis xerxebiT.
a) triangulaciis meTodis arsi SemdegSi mdgomareobs: adgilze qmnian
samkuTxedebis jaWvs ise, rom wveroebidan erTmaneTi Candes. A da B
punqtebis koordinatebi cnobilia (gansazRvrulia astronomiulad da
maTematikurad gamoTvlilia geodeziuri koordinatebi), izomeba Siga
kuTxeebi 1, 2, 3, 4... cnobili A da B wertilebis koordinatebisa da gazomili
kuTxeebis saSualebiT gamoiTvleba danarCeni punqtebis koordinatebi.
mocemuli A wertilis koordinatebiT, AB xazis sigrZiT, misi (AB)
azimutiT da gazomili Siga kuTxeebiT 1, 2, 3, 4, ..., gamoiTvleba BCDEF ...
wertilebis koordinatebi (nax. 15).
b) trilateraciis dros gamosavali monacemebia igive e.i. 1) A da B
wertilebis koordinatebi an 2) A wertilis koordinatebi, AB-s sigrZe da
(AB)-azimuti. nacvlad kuTxeebisa izomeba yvela gverdis sigrZeebi, xolo
Semdeg gamoiTvleba danarCeni wertilebis koordinatebi.
g) poligonometriis dros A wertilia cnobili da viciT pirveli AB
gverdis azimuti. vzomavT S1, S2, S3. . . manZilebs da kuTxeebs 1β , 2β , 3β . . .
Semdeg gamoiTvleba B, C, D. . . wertilebis koordinatebi (nax. 16).
2
AA
((AABB)) SS11
BB
β 1
β 2
β 3
SS SS
CC
DD
nax. 16
2 3
3366
d) niveloba
geodeziuri dasayrdeni qselis sasimaRlo safuZvlis Seqmna xdeba,
geometriuli, trigonometriuli da barometruli nivelobis xerxiT.
niveloba aris moqmedebaTa erToblioba, romlis saSualebiTac
ganisazRvreba wertilTa Soris simaRleTa sxvaoba, anu aRmateba.
geometriuli nivelobis dros xdeba wertilebs Soris simaRleTa
sxvaobis gansazRvra Tarazuli sxivis saSualebiT, romelsac gvaZlevs
xelsawyo (niveliri).
17-e naxazidan Cans, rom simaRleTa sxvaobis gansazRvris formulaa
bah −= . (4.1.1)
trigonometriuli niveloba sruldeba daxrili sxivis saSualebiT,
sadac daxris kuTxe da manZilebi izomeba TeodolitiT, romelTa
monacemebis CasmiT saTanado formulebSi miiReba wertilTa Soris
simaRleTa sxvaobebi.
barometruli niveloba sruldeba wertilebze haeris wnevisa da
temperaturis gazomvebis Sedegad.
4.2. gegmebis, rukebis, profilebis SedgenasTan
dakavSirebuli zogierTi cnebebi
nebismieri obieqtis geometriuli ageba advilia mis msgavs zedapirze.
nax. 17
3377
maSasadame, dedamiwis namdvil saxeze srulyofili warmodgena gveqneba,
Tu mas avagebT mis msgavs tanze, magaliTad – globusze, magram praqtikuli
gamoyenebisaTvis didi globusi uxerxulia (magaliTad 1 : 10 000 000
masStabSi dedamiwa globusze gamoisaxeba 1.3m diametriT. masze everestic ki
Znelad SeimCneva, amitom igi naklebad gamosayenebelia), ris gamoc dedamiwas
an mis did nawilebs gamoxazaven sibrtyeze (qaRaldze) gegmebis an rukebis
saxiT.
sferos ganSla sibrtyeze dakavSirebulia damaxinjebebTan (dedamiwa
rom iyos cilindruli an konusuri formis igi daumaxinjeblad
gaiSleboda), amitom geodeziaSi miRebulia sxvadasxva kartografiuli
proeqciebi, sadac mxedvelobaSi miRebulia zemoxsenebuli damaxinjebebi. aq
saqme gveqneba or SemTxvevasTan:
1. roca sivrce didia;
2. roca sivrce pataraa.
roca sivrce ar aRemateba 300 – 320 km2, saqme gveqneba gegmebTan, xolo
roca sivrce metia, miviRebT rukebs.
patara sivrces vagegmilebT Q horizontul sibrtyeze orTogona-
lurad. aq Sveuli xazebi paralelurebi iqnebian. (nax. 18)
didi sivrceebi gegmildeba sferul zedapirze, sadac Sveuli xazebi
urTierT mimarT paralelurebi ar iqnebian. (nax. 19)
A0
E0 D0
C0
B0
B
C
D E
A
gegma 1 : 5000
c a
e d
b
nax. 18 Q
3388
orive SemTxvevaSi adgili aqvs damaxinjebas, magram pirvelSi
gansxvaveba mcirea da naxazze gadagvaqvs saTanado SemcirebiT, gegmis
saxiT, sadac damaxinjebis Secdomas vugulvebelvyofT.
pirvel SemTxvevaSi A0B0C0D0E0 uwodeben Tarazul proeqcias
(horizontul proeqcias).
meore SemTxvevaSi ki uwodeben horizontalur proeqcias.
dedamiwis fizikur zedapirs warmodgeniT vagegmilebT donebriv
zedapirze, rac saerTod ar xdeba. sinamdvileSi ki, AB xazis qvedebuli es
aris AB xazis proeqcia sibrtyeze (nax. 20).
nax. 20
nax. 19
A0
E0 D0
C0 B0
C
D E
A qvedebuli
B
3399
nax. 21
2211'0
'"0
"0
'00 coscos νν LLLLLLL ++=++= (4.2.1)
maSasadame, adgilze Cven vzomavT an vsazRvravT qvedebuls. mcire
sivrceebze qvedebulebi da Tarazuli proeqciebi erTi da igivea, xolo did
sivrceebze qvedebulebSi unda SevitanoT Sesworebebi.
gegma aris sibrtyed (Tarazuli) miRebul donebriv zedapirze
orTogonalurad dagegmilebuli, SedarebiT mcire teritoriis asli,
gamoxazuli qaRaldze.
didi teritoriis dros hori-
zontaluri proeqciidan aviRoT qvede-
bulebi (nax. 22).
HRR
ABBA
+='
0
00 (4.2.2)
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅⋅−+−=
+= 2
2'0
'000 1
RH
RHAB
HRRABBA (4.2.3)
am gamosaxulebaSi bolo wevrs simciris gamo ugulvebelvyofT, amitom
⎟⎠⎞
⎜⎝⎛ −≈
RHABBA 1'
000 . (4.2.4)
Sesworeba gamoiTvleba formuliT
RHABABBA '
0'000 −=−=ε . (4.2.5)
qvedebuli
proeqcia
mcire terito-riisaTvis
A
B
A0 B0
B0'ν
A B
H
B0 A0
B'
B0'
didi sivrce-ebisaTvis
Z. d. z.
nax. 22
4400
magaliTad, roca 500'0 =AB m, maSin gamoTvlis Sedegad 8=ε sm.
vinaidan horizontaluri proeqcia qvedebuls ar udris, amitom
sferoidis zedapirze dagegmilebis dros qvedebuli maxinjdeba. e.i. rukis
Sedgenis dros qvedebulebi orjer unda Seswordes.
ruka aris horizontalurad miRebuli sferoidul an sferos
zedapirze orTogonalurad dagegmilebisa da sibrtyeze gamoxazvis dros
damaxinjebisagan ara srulad Sesworebuli Semcirebuli gamosaxuleba
mTeli dedamiwisa an misi didi nawilebisa qaRaldze.
gegma da ruka maSin mogvcems srulyofil warmodgenas dedamiwis
fizikur zedapirze, Tu orive SemTxvevaSi gvecodineba yvela konturuli
wertilebis niSnulebi.
4.3. cnebebi agegmvebis Sesaxeb
agegmvis qveS vgulisxmobT moqmedebaTa erTobliobas, romlis
saSualebiTac Seiqmneba raime nagebobis gegma da Wrilebi, adgilis
topografiuli gegma da rukebi, an arsebul gegmebSi da rukebSi sxvadasxva
daniSnulebisamebr Seitaneba saWiro wvliladebi.
agegmviTi samuSaoebi SeiZleba daiyos 2-jgufad: 1. savele da 2.
kameraluri.
savele samuSaoebis qveS igulisxmeba manZilebisa da kuTxeebis
gazomvebi.
kameraluri samuSaoebi iTvaliswinebs ganazomTa maTematikur
damuSavebas, gamoTvlebs da gamoxazvas.
agegmvebi sruldeba: 1. dedamiwis fizikur zedapirze, 2. wiaRSi, 3.
sahaero agegmvebi.
dedamiwis fizikur zedapirze agegmvebs ekuTvnis agegmvebi xmeleTze.
saxmeleTo agegmvebia: instrumentaluri, naxevrad instrumentaluri da
TvalzomiTi.
xSirad agegmvis saxelwodeba dakavSirebulia instrumentTan e.i. im
instrumentis saxelwodeba hqvia, romliTac xdeba agegmva. magaliTad:
Teodolituri, menzuliTi, taxeometriuli, fotoTeodolituri da sxva.
wiaRis agegmvebi gankuTvnilia: a) metropolitenebisaTvis da b) samTo-
saeqsploatacio saqmesTan dakavSirebuli.
4411
metropolitenebisaTvis gankuTvnili samuSaoebia: saorientiro,
poligonometriuli samuSaoebi da niveloba.
samTo-saeqsploatacio saqmesTan dakavSirebul samuSaoebSi, anu
samarkSeidero samuSaoebSi Sedis: saorientiro, Teodolituri, niveloba da
wvliladebis agegmvebi.
aeroagegmva warmoadgens aerogadaRebis, savele laboratoriuli
samuSaoebis, topogeodeziuri samuSaoebis da fotogrammetriuli
samuSaoebis erTobliobas.
yovel agegmvaSi ZiriTadia zogadidan kerZoze gadasvlisa da
aucileblobisa da sakmarisobis principis dacva.
arsebul rukebSi wvliladebis SetanasTan dakavSirebuli agegmvebi
sruldeba sxvadasxva samecniero dargebis mier wvliladebis Setanis
mizniT, aRmoCenebis Setanis mizniT (magaliTad, agegmvebi arsebobs
geologiuri, geofizikuri, sagzao, satyeo, sinoptikuri, sahaero, eleqtruli
da sxva), ris Sedegad miiReba saTanado saxelwodebis gegmebi, rukebi da
profilebi.
4.4. masStabebi
gegmebis, profilebis, rukebis, aseve nebismieri naxazis qaRaldze
gamoxazvisaTvis saWiroa Semcireba, rasac masStabs vuwodebT. ZiriTadad
masStabebia: vertikaluri da horizontaluri.
vertikalur masStabebs (profilebis Sedgenis dros) vuwodebT
niSnulebis Sesabamisi simaRleebis, profilebis Semcirebis zomas.
gegmaze horizontuli masStabia qvedebulebis Sesabamisi sigrZeebis
qaRaldze gamoxazvis Semcirebis zoma.
mocemul wertilSi, garkveuli mimarTulebiT rukis masStabebi
ewodeba am wertilis maxloblobaSi imave mimarTulebiT aRebul
usasrulod mcire sigrZesa da adgilze Sesabamis Sesworebul qvedebulebs
Soris fardobas.
a) ricxviTi masStabi
4422
ricxviTi (1:M)masStabi ewodeba aliqvoturad (mricxvelSi erTi)
daweril wilads, romlis mniSvneli gviCvenebs ramdenjer unda Semcirdes
qvedebuli sibrtyeze gamoxazvis dros: 10000
1,
50001
, 2000
1.
mocemuli ricxviTi masStabebi SegviZlia davweroT sxvagvaradac:
1sm – 100m; 1sm – 50m; 1sm – 20m. ricxviTi masStabis safuZvelze ase daweril
masStabs ewodeba saxeldebuli (ganmartebiTi) masStabi.
masStabebis cnebasTan dakavSirebiT SeiZleba amoixsnas sami amocana.
1. gvaqvs gegma, romlis masStabi ar viciT da unda gavigoT am gegmis
masStabi.
vTqvaT, xazis monakveTis sigrZe gegmaze 2sm-ia, xolo am monakveTis
Sesabamisi sigrZe adgilze=200m.
maTi SefardebiT miviRebT saZiebel masStabs.
100001
200002
= ;
2. gvaqvs xazis sigrZe gegmaze da masStabi. unda gavigoT Sesabamisi
qvedebuli adgilze. xazis sigrZe gegmaze gamravlebuli masStabis
mniSvnelze iqneba qvedebulis sigrZe adgilze.
3. gvaqvs qvedebulis sigrZe adgilze da masStabi. qvedebulis
sigrZes gavyobT masStabis mniSvnelze, miviRebT monakveTis qvedebulis
sigrZes gegmaze.
vinaidan zemoxsenebuli ariTmetikuli moqmedebebi, fargliT
gamonaTvlis aReba saxazavze da misi gadazomva qaRaldze bevr dros
moiTxovs da Secdomebis wyaroa, amitom mimarTaven xazviT masStabs.
b) xazviTi masStabi
xazviTi masStabi cnobilia Semdegi saxis: xazovani, ganivi, soluri.
nebismieri saxis xazviTi masStabi (xazovani, ganivi, solisebri)
warmoadgens skalas, romlis danayofis kvesurebze warwerilia mocemuli
ricxviTi masStabis Sesabamisi ricxvebi.
xazovani masStabis Sesadgenad farglis maqsimaluri bijiT (aTi
santimetri) gavatarebT Tarazul sqel xazs, Semdeg mis paralelurad erT
4433
mm-ze daSorebiT gavatarebT bewvisebur xazs (ara umetes 0.08 mm-sa),
romelsac zomis xazi ewodeba.
mocemuli ricxviTi masStabis safuZvelze SevarCevT masStabis fuZes
(Cvens SemTxvevaSi 1sm) saerTod iReben 1sm, 2sm, 2.5sm. e.i. iseT fuZes,
romelsac mocemuli ricxviTi masStabis mixedviT Seesabameba martivi
ricxvebi. magaliTad 1 : 10000, 1 : 5000, 1 : 2000 masStabisaTvis Sesabamisad
fuZes aviRebT 1, 2 da 2.5sm. Semdeg gadavzomavT gavlebul xazze SerCeuli
fuZis tol monakveTebs.
vTqvaT, ricxviTi masStabi aris 1 : 10000, maSin xazovani masStabi iqneba
(nax. 23)
nax. 23
a monakveTs masStabis fuZe ewodeba, xolo adgilze mis Sesabamis
qvedebuls ki masStabis odenoba hqvia. Tu a=2sm, masStabs normuls
uwodeben.
fuZis pirvel marjvena kvesurze wavawerT 0-s, marcxniv mocemuli
ricxviTi masStabis Sesabamis ricxvs adgilze Seesabameba – 100m. pirvel
fuZes vyobT m=10 tol nawilad.
masStabis sizuste ewodeba umciresi danayofis Sesabamis qvedebuls
adgilze. igi cvalebadi sididea erTi da igive masStabisaTvis da
damokidebulia masStabis agebaze (magaliTad Tu fuZes davyobT 10 nawilad,
Cveni masStabis SemTxvevaSi sizuste iqneba 10m, Tu davyobT 20 nawilad –
sizuste iqneba 5m da xuT nawilad dayofis SemTxvevaSi sizuste iqneba 20m).
masStabis zRvruli sizuste aris mocemuli masStabis zRvruli
grafikuli sizustis Sesabamisi qvedebuli adgilze.
zRvruli grafikuli sizuste yvela masStabisaTvis mudmivia.
magaliTad: aviRoT wertili da davmziroT saukeTeso xedvis 250=ω mm
manZilze ε kritikuli kuTxiT. eqsperimentalurad dadgenilia rom
igi "" 6057 −= .
1.0250206265
60"
"
"
≈⋅=⋅= ωρεd mm
e.i. wertilis diametri, romelsac arCevs SeuiaraRebeli Tvali
yofila 0.1mm, rasac ewodeba masStabis zRvruli grafikuli sizuste.
dω
"" 6057 −=ε -
100 0 100 200 300 400
aa
4444
1 : 10000 masStabisaTvis zRvruli sizuste 1m iqneba.
1 : 5000 masStabisaTvis zRvruli sizuste 50sm iqneba.
1 : 2000 masStabisaTvis zRvruli sizuste 20sm iqneba.
rogorc vxedavT yvela masStabisaTvis zRvruli grafikuli sizuste
erTi da igive iqneba, xolo zRvruli sizuste ki sxvadasxva.
anaTvali xazovan masStabze udris farglis marjvena da marcxena
wveroze anaTvlebis jams, magaliTad: vTqvaT 1 : 10000 masStabis saSualebiT
Sedgenil xazovan masStabze unda gadavzomoT 271.35m, mocemul masStabSi 1m-
ze nakleb sigrZes ver gadavzomavT e.i. 0.35 metri unda gadavagdoT. 200m
gadavzomavT marjvniv da 70m-is gadasazomad marcxena fexs gadavitanT
fuZis me-7 patara danayofze; 1m-is gadasazomad ki imave marcxena fexs
gadavitanT kidev marcxniv milimetris 0.1-ze. gazomili xazis sigrZe iqneba
200+70+1=271m.
g) ganivi anu transversaluri masStabi
ufro meti sizustis saWiroebis SemTxvevaSi gamoiyeneba ganivi anu
transversaluri masStabi. SevirCevT fuZes da xazovani masStabis principze
avagebT masStabs e.i. gadavzomavT fuZeebs da gadazomvis wertilebSi
avmarTavT perpendikularebs, gavukeTebT warwerebs 1, 0, 1, 2, 3, ...
vTqvaT fuZed aviReT 2sm. pirveli monakveTi davyoT m=10 nawilad.
SevarCioT monakveTi 2-3mm da ganapira marTobebze gadavzomavT n=10
monakveTs.
nax. 24
pirveli zeda fuZe gavyoT 10 nawilad da D wertili SevuerToT
qveda fuZis mecxre kvesurs. Semdeg gavavlebT D9 xazis paralelur xazebs
anu transversalebs.
D A B
8 6 4 2 0
8 6 4 2
0 1 2 3 4
P1
P2
a1
a2
b1
b2
aq bq a2
a1
*
*
*
*
4455
fuZis patara danayofi udris mis 0.1-s, xolo vertikalurad
gadaadgileba TiTo safexuriT iqneba fuZis patara danayofis 0.1-s an
fuZis 0.01-s.
e.i. 01.011 =ba fuZisas
02.022 =ba fuZisas
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
0.1AB = fuZisas
vTqvaT, unda gadavzomoT 286m 72sm (masStabis fuZea erTi santimetri)
fuZis jerad ricxvs 200m gadavzomavT fuZis marjvniv 2-Tan, e.i. farglis
marjvena fexs gavaCerebT 2-ze. Semdeg 80m-is gadasazomad farglis
marcxena fexs davayenebT fuZis me-8 danayofze. Semdegi ricxvis – 6m-is
gadasazomad farglis marcxena fexs gadavaadgilebT vertikalurad zeviT
me-6 danayofze. Semdegi ricxvis 70sm-is gadasazomad farglis fexs
gadavaadgilevT vertikalurad zeviT me-6 danayofidan 0.7 monakveTze
(TvaliT), xolo ukanasknel ricxvs – 2sm gadavagdebT.
amrigad, Cveni gadazomili sigrZe iqneba 2.867sm. maSasadame, ganivi
masStabi 10-jer ufro zustia xazovan masStabze da gamosayeneblad
moxerxebulia.
masStabi wilad fuZeze
im SemTxvevaSi, rodesac ricxviTi masStabis mniSvneli ar aris
gamosayeneblad moxerxebuli ricxvi an vsargeblobT arametruli aTobiTi
sistemis masStabiani rukebiT da gegmebiT, muSaobis
gaadvilebis mizniT adgenen iseT xazviT masStabs,
romelsac fuZe eqneba wiladi, mxolod qvedebulze
Sesabamisi ricxvebi mrgvali iqneba. magaliTad,
solisebri masStabis fuZea (nax. 25) 1,11 sm, imis gamo,
rom aerosuraTis ricxviTi masStabis mniSvneli
davamrgvaleT Semdegnairad: aerosuraTis masStabi
aris 1 : 18000, e.i. gegmaze 1 sm qvedebuls adgilze
Seesabameba 180 m. vadgenT proporcias – ras unda
nax. 25
4466
udrides a fuZe, rom Sesabamisi qvedebulis sigrZe adgilze 200 metri iyos,
maSasadame,
1 sm ---------- 180 m
a --------- 200 m
an kidev, vTqvaT, vsargeblobT 200 sJ rukiT, e.i. gegmaze 1 gojs adgilze
Seesabameba 200 sJ. saerTod, pirvel rigSi unda gavigoT ricxviTi masStabi,
risTvisac versebi da saJenebi gojebSi unda gamovsaxoT. maSasadame,
vinaidan 1 sJ=84 gojs, 200 sJ rukis masStabi iqneba 1 : 16800. vadgenT
proporcias.
mocemuli 200 saJeniani rukis masStabis mixedviT 1 sm fuZes adgilze
qvedebuli Seesabameba 168 metri. ras unda udrides a fuZe, rom mas
Seesabamebodes 200 metri, e.i. masStabis fuZe unda iqnes
a =⋅
=200 11904m 1 sm
168 m sm,
ganxiladi rukisaTvis nebismieri saxis xazviTi (xazovani, solisebri, ganivi)
masStabis fuZes aviRebT 1,19 an 2, 38 sm. pirvel SemTxvevaSi adgilis
qvedebulis Sesabamisi ricxvebi iqneba 200 metri da meore SemTxvevaSi 400
metri. amiT rukis ricxviTi masStabi igive iqneba da asec unda iyos, xolo
wiladfuZiani masStabis sizuste mcireoden Semcirdeba.
saerTod unda gvaxsovdes, xazviTi masStabis fuZe unda aviRoT 1,19 an
2,38 sm. pirvel SemTxvevaSi saTanado ricxvebi adgilze iqneba imdeni metri,
ramden saJeniani rukac gvaqvs, xolo meore SemTxvevaSi metrebis raodenoba
orjer meti iqneba, magaliTad, 500 sJ rukisaTvis (rukaze 1 gojs adgilze
qvedebuli Seesabameba 500 sJ. ricxviTi masStabi ki iqneba 1 : 42000) xazviTi
masStabis 1,19 fuZes adgilze 500 metri qvedebuli Seesabameba, xolo 2,38 sm
fuZes ki adgilze qvedebuli Seesabameba 1000 metri. ricxviTi masStabi igive
darCeba. masStabs wilad fuZeze xSirad gadasayvani masStabi ewodeba. am
RonisZiebiT nebismieri sistemiT Sedgenil rukebze ganazomebi metrul
sistemaSi miiReba, rac metad uwyobs xels Zveli (saJenuri) da sxvadasxva
sistemis rukebis gamoyenebas. arsebuli ganivi masStabidan SeiZleba aviRoT
gadasayvani masStabebis fuZeebi Semdegnairad:
magaliTi: vTqvaT, masStabi aris erTsantimetriani fuZiT, maSin
2 2 1,19Q P = sm, xolo 3 3 2,38Q P = sm.
a =⋅
=200 1111m 1sm
180m sm,
4477
magaliTi: vTqvaT, masStabi aris orsantimetriani fuZiT, maSin
4 4 1,19Q P = , xolo 5 5 2,38Q P = sm.
4.5. rukebisa da gegmebis klasifikacia
rukebze da gegmebze moTavsebul adgilis Sesaxeb cnobebis
erTobliobas misi Sinaarsi ewodeba. Sinaarsis mixedviT rukebi gvaqvs
1. saerTo geografiuli da 2. specialuri
masStabis mixedviT saerTo geografiuli rukebi aris sami saxis:
1. wvrilmasStabiani, romlis mniSvnelia 1000000 – mimoxilviTi rukebi.
2. saSualo masStabiani mimoxilviTi rukebi:
1 : 1000000, 1 : 500000, 1 : 300000
3. msxvilmasStabiani topografiuli rukebi
1 : 200000, 1 : 100000, 1 : 25000
4. topo – gegmebi
1 : 10000, 1 : 5000, 1 : 2000, 1 : 1000 da 1 : 500
specialuri ewodeba iseT rukebs, romlebzec ufro detalurad aris
gamosaxuli saerTo geografiuli rukebis calkeuli elementebis Sinaarsi
an gamosaxulia iseTi bunebrivi da socialur-ekonomikuri elementebi,
romlebic saerTo geografiul rukebze ar aris gadaRebuli. magaliTad,
specialur rukebs ganekuTneba geologiuri, geofizikuri, satyeo, sagzao,
sazRvao, sahaero, gazuri, eleqtruli, sinoptikuri da sxva (maT xSirad
Tematur rukebsac uwodeben). samxedro rukebi, specialur rukebs ekuTvnis,
romlebzec gadatanilia reliefisa da obieqtebis garda, adgilze
transportisa da sxva samxedro aRWurvilobis gamtarunarianobis
(gruntebi, niadagebi) da svlebis Sesaxeb cnobebi da sxva. samxedro
rukebia: strategiuli (1 : 1000 000 da ufro wvril masStabiani rukebi),
operatiuli (1 : 500 000 – 1 : 200 000 masStabiani), taqtikuri (1 : 100 000 – 1 :
25 000 masStabiani). saerTod ki unda vicodeT, rom rukebi da gegmebi erT-
erTi umniSvnelovanesi masala da dasayrdenia samxedro saqmis warmoebis
dros. amitom amboben, rom ruka „armiis Tvaliao“. saerTod ki Znelia
warmovidginoT iseTi dargi, romelic ar saWiroebs rukebs, gegmebs da
profilebs.
4488
nax. 26
4499
4.6. rukebis dayofa, aRniSvna – danomrva
da nomenklatura
praqtikuli saWiroebis Sesabamisad mTeli dedamiwa gamoisaxeba
rukebis calkeul furclebze, risTvisac saWiroa mTeli dedamiwis dayofa.
dayofisaTvis gamosavlad miRebulia 1 : 1000000 ruka.
ekvatoridan zeviT da qveviT dedamiwas yofen ο4 -ian sferul
sartylebad – sul 44, romelsac nomraven laTinuri mTavruli asoebiT A, B,
C, . . . V-mde
amave dros grinviCis meridianis ο180 -dan gamoyofen ο6 -ian sferul
orgverdovanebs meridianebis saSualebiT, rasac fiuzos uwodeben da
nomraven 1, 2, 3, 4, . . . , 60. mTeli dedamiwa daiyofa 44 60 2640× = trapeciad.
unda vicodeT, rom ama Tu im masStabiani rukis yovel furcels
asoebiT da ricxvebiT Tavisi aRniSvnebi an nomrebi aqvs. rukis furclebis
aRniSvnebisa da danomvris sistemas nomenklatura ewodeba.
vTqvaT gvinda ganvixiloT trapecia, romelSic Tbilisi mdebareobs,
misi nomenklaturaa K-38, sadac K-sartylis nomeria, 38-fiuzos nomeria.
5500
a)
b)
nax. 27
c)
d)
pirvel cxrilSi ganxilulia sxvadasxva masStabiani rukebis da
gegmebis nomenklatura Tbilisis (K-38) sartylisaTvis•.
•• amJamad saerTaSoriso asociaciis mier miRebulia Tanamedrove nomeklatura. radganac jerjerobiT iyeneben zemoaRniSnul nomenklaturas, amitomac vawvdiT mkiTxvels am masalas.
47 45'o
a
c d
b
K-38-22-B47 45'o
4330 '
o 4330 '
o
4320 '
o 4320 '
o
48 00 'o
4325'
o
48 00 'o
1: 50 000
4325'
o
dd
1
3 4
2
K-38-22-B-d47 45'o
4325'
o 4325'
o
4320 '
o 4320 '
o
47 52 '30"o
4322 '30"
o
47 52 '30"o47 45'o 1: 25 000
44
K-38-22-B-d-447 58'7.5"o
432 '15"
o 432 '15"
o
4320 '
o 4320 '
o
48 00 'o
4322 '37,"5
o
48 00 'o47 58'7,"5o 1: 5 000
A
C D
B a
c d b
K-38-2247 30 'o
4340 '
o 4340 '
o
4320 '
o 4320 '
o
48 00 'o
1:100 000 47 30 'o 48 00 'o
38
A B C
K
N
S
A B D C
1 2 3 4 5 6 7 8 9 10 11 12 I II III IV V VI
VVIIII 2233
XXXXXXVVII 140 141 142 143 144
4ο
6ο
1133
48 00 'o42 00 'o
4400 '
o 4400 '
o
4000 '
o 4000 '
o
48 00 'o42 00 'o 1:1 000 000
5511
cxrili 1
rkali
masStabebi gamosavali
furceli dayofa
aRniSvna
an dayofanomenklatura
paral
eli
merid
iani
1 : 1000000
1 :500000
1 : 300000
1 :200000
1 : 100000
1 : 50000
1 : 25000
1 : 10000
1 : 5000
1 : 2000
1 : 1000
1 : 500
dedamiw. zed
381000000:1
−K
381000000:1
−K
“
“
22381000000:1
−−K
BK −−− 223850000:1
dBK −−−− 223825000:1
2238100000:1
−−K
)240(22385000:1
−−−K
K −−−− 24022382000:1
K −−−− 24022382000:1
2640
4
9
36
144
4
4
4
256
4
4
16
A, B, C, . . .
1, 2, 3, . . .60
А, Б, С, Д,
I-II-III. . .IX
I-II-. .
.XXXVI
1, 2, 3, . . .144
А, Б, С, Д,
a, b, c, d
1, 2, -3,4
1, 2, 3, . . .256
A, B, C, D
I-II-III-IV
1, 2, 3, . . .16
K – 38
K – 38 – A
K – 38 – VII
K – 38 – VII
K – 38 – 22
K – 38 – 22 – B aqedan iwyeba sibrtye
K – 38 –22 – B – d
K – 38 – 22 – B – d –2
K – 38 – 22 – (240)
K – 38 – 22 – 240 – A
K–38– 22 –240–A–III
K–38– 22 –240–A–12 K – 38 – 22 – 240 – A
ο6
ο3
ο2
ο1
'30
57',
453',
ο4
ο2
201ο
'40
'20
'5
52',
mimoxilviTi wvrilmasStabiani rukebi
topografiuli rukebi
topografiuli gegmebi
5522
sakiTxis SedarebiT srulad warmodgenisaTvis saWirod migvaCnia
moviyvanoT saqarTvelos teritoriisaTvis amJamad dadgenili topog-
rafiuli rukebis Semdegi masStaburi rigi da trapeciebis CarCoebis zomebi,
aseve miRebulia Sesabamisi nomenklatura.
moviyvanoT cxrili prezidentis 1999 wlis #206 brZanebulebidan.
BΔ _ ganedebis sxvaoba
LΔ _ grZedebis sxvaoba.
cxrili 2
trapeciis CarCos zomebi # masStabi
BΔ LΔ smXsm nomenklatura
1
2
3
4
5
6
7
8
1 : 1 000 000
1 : 500 000
1 : 200 000
1 : 100 000
1 : 50 000
1 : 25 000
1 : 10 000
1 : 5 000
4 00 'o
2 00 'o
0 48'o
0 24 'o
0 12 'o
0 06 'o
0 03'o
0 01'30"o
6 00 'o
3 00 'o
112 'o
0 40 'o
0 20 'o
0 10 'o
0 05'o
0 02 '30"o
44.4x49.7
44.4x49.7
44.4x49.7
44.4x55.2
44.4x55.2
44.4x55.2
55.5x69.0
55.5x69.0
T-38
T-38-E
T-38-XLII
T-38-C4804
T-38-L4406
T-38-4635
T-38-4635d
T-38-4635d-4
5533
V Tavi
wertilTa mdebareobis gansazRvris sakiTxisaTvis
5.1. orientireba
a) mimarTulebis dadgenisaTvis
nax. 28
sivrceSi C wertilze mimarTulebis, anu CDn mimarTulebis
gansazRvrisaTvis, saWiroa Tarazuli mimarTuleba anu ϕ kuTxe da
vertikaluri mimarTuleba anu ν kuTxe.
ϕ aris Tarazuli polaruli kuTxe, romelsac ewodeba Tarazuli
mimarTuleba, romelsac mimarTuli anu 0nCD qvedebuli qmnis C wertilSi
gatarebul x RerZTan (RerZi ki mimarTuli xazia).
kuTxe ν aris daxris kuTxe, romelsac adgens mimarTuli nCD
monakveTi mis 0nCD qvedebulTan.
orientireba ewodeba mxolod ϕ kuTxis gansazRvras, ris Sedegadac
C wertili moxvdeba 0nCD qvedebulis gaswvrivobaSi.
orientirebasTan dakavSirebiT ZiriTadad sami amocana wydeba:
1. TviT ϕ kuTxis gansazRvra;
2. cnobili (gegmidan amoRebuli) ϕ kuTxis saSualebiT D wertilis
gagneba;
5544
3. arsebuli rukebisa da gegmebis orientireba, rac niSnavs rukebisa
da gegmebis ise Semobrunebas, rom maTze moTavsebuli obieqtebi
moxvdnen adgilze maTi Sesabamisi obieqtebis gaswvrivobaSi.
x RerZis nacvlad C wertilSi SeiZleba gvqondes astronomiuli
meridianis, geodeziuri meridianis, RerZa meridianis an magnituri
meridianis proeqciebi, sferoidul, sferul an donebriv sibrtyeze, ris
Sesabamisad saorientiro kuTxeebs ewodebaT astronomiuli azimuti an
rumbi, geodeziuri azimuti an rumbi, direqciuli kuTxe an direqciuli rumbi,
magnituri azimuti an magnituri rumbi.
saugulvebelyofo Secdomis daSvebiT astronomiul da geodeziur
meridianebs Serwymulad sTvlian (sinamdvileSi isini urTierT gadaxrilebi
arian) da orives erTad uwodeben WeSmarit an geografiul meridians.
aqedan saTanado (astronomiul da geodeziur) azimutebs, saerTod
uwodeben WeSmarit an geografiul azimuts da maSasadame orientirebasac
uwodeben WeSmarit an magnitur orientirebas. aqedan gamomdinare
orientirebac orgvaria: geografiuli da magnituri.
b) WeSmariti orientireba
nax. 29
vTqvaT NS aris O wertilis geografiuli meridiani e.i. sadac
mixriloba mxedvelobaSi ar aris miRebuli.
5555
wertilebze orientireba xdeba kuTxeebiT: 1) azimuti 2) direqciuli
kuTxe da 3) rumbi. WeSmarit orientirebisas viyenebT WeSmarit azimuts,
direqciul kuTxes an WeSmarit rumbs.
NS aris saSuadReo xazi (proeqcia donebriv zedapirze) WeSmariti an
geografiuli meridiani. OM1 mimarTulebiT monakveTis WeSmariti azimutia
A1.
azimuti aris kuTxe (A), romelsac OM1 – mimarTuli qvedebuli qmnis O
wertilis WeSmarit saSuadReo xazis Crdilo mimarTulebasTan. icvleba
ο0 -dan ο360 -mde saaTis isris moZraobis mimarTulebiT. (nax. 37)-dan.
OM1 – azimuti iqneba A1
OM2 – azimuti iqneba A2
OM3 – azimuti iqneba A3
OM4 – azimuti iqneba A4
g) damokidebuleba pirdapir da Sebrunebul
azimutebs Soris
azimutebi aris: 1) pirdapiri (romelsac gvaZleven an vzomavT) da 2)
Sebrunebuli, romelic gamoiTvleba (nax. 30)
nax. 30
vTqvaT gvaqvs CD xazis O wertilSi pirdapiri azimuti Ap, maSin imave
wertilSi misi Sebrunebulia AS. naxazidan
5566
A AS p= ± 180ο
wrfis erTi da igive wertilSi Sebrunebuli azimuti udris pirdapir
azimuts ±180ο , (+ roca pirdapiri azimuti <180ο da _ roca pirdapiri
azimuti >180o) (nax. 30)
d) wrfis azimutis qceva
nax. 31
aviRoT mimarTeba CD. O1 wertilSi misi azimuti iqneba A1, O2-Si ki A2
(aq laparakia WeSmarit orientirebaze e.i. gveqneba WeSmariti azimuti).
CD xazze sxvadasxva wertilSi azimuti sxvadasxvaa. igi izrdeba
aRmosavleTiT da mcirdeba dasavleTiT.
γ+= 12 AA ,
γ -meridianis Seaxloebis kuTxe.
5577
e) wrfis sxvadasxva wertilSi azimutebs
Soris damokidebuleba
aviRoT igive CD xazi. vTqvaT O1 wertilSi pirdapiri azimutia Ap .. O2
wertilSi imave CD-s Sebrunebuli azimuti AS . wrfis sxavadasxva wertilSi
Sebrunebuli azimuti tolia
180A A γ= ± ±oS p .
pliusia, roca p 180A < o da minusi ki roca p 180A > o
nax. 32
5.2. direqciuli kuTxe
cneba direqciuli kuTxis Sesaxeb CvenSi Semovida mas Semdeg, rac
miRebuli iqna gausis proeqciebi da koordinatebi.
aviRoT sferuli orgverdovanedi (fiuzo). TviToeul fiuzoSi
centralur meridians uwodeben RerZa meridians, romelsac x RerZs
amTxveven. RerZa meridianis marTobi aris ekvatoris proeqcia da
aRniSnaven y-iT. fiuzos farglebSi meridianebs Tvlian urTier-
Tparalelurad. paralelebis proeqciebic iqnebian paralelurebi.
5588
paralelebisa da meridianebis gavlenis Sedegad miiReba
kilometruli bade (nax. 33)
nax. 33
direqciuli kuTxe ewodeba im kuTxes, romelsac mimarTuli qvedebuli
qmnis RerZa meridianis paraleluri RerZis Crdilo mimarTulebasTan.
icvleba 0ο -dan 360ο -mde saaTis isris moZraobis Tanxvdenilad da
aRiniSneba α asoTi; maSin roca wrfis wertili moxvdeba RerZa meridianze
– direqciuli kuTxe emTxveva azimuts. wrfis azimuti icvleba, wrfis
direqciuli kuTxe ki mudmivia.
a) damokidebuleba wrfis sxvadasxva wertilSi azimutebsa
da direqciul kuTxes Soris
Tu gavatarebT OM1-is romelime wertilSi WeSmarit meridians,
miviRebT azimuts. davukavSiroT azimuti direqciul kuTxes (nax. 33).
γα += 11A , (5.2.1)
es maSin, rodesac wertili RerZa meridianis aRmosavleTisakenaa, xolo
γα −= 22A , (5.2.2)
5599
roca wertilia RerZa meridianis dasavleTiT. magaliTad OM2-is
romelime wertilSi 2α -iqneba direqciuli kuTxe, aqve gavataroT WeSmariti
meridiani, miviRebT 2γ kuTxes
e. i. γα −= 22A
maSasadame, zogadad orive SemTxvevisaTvis davwerT
γαγα +=±=A (5.2.3)
e.i. WeSmariti (geografiuli) azimuti udris direqciuli kuTxisa da
meridianTa Seaxloebis algebrul jams (radganac γ -Seaxloeba algebruli
ricxvia)
5.3. meridianTa Seaxloebis kuTxis
gamosaTvleli formula
CN da DN arian C da D wertilebis saSuadReo xazebi. maT Soris
kuTxe γ iqneba meridianTa Seaxloeba.
CD LLL −=Δ , (5.3.1)
anu LΔ aris C da D wertilebis grZedebis sxvaoba.
nax. 34
sferoze or C da D wertilebs Soris rkali CD=S. meridianTa
Seaxloebis gansazRvrisaTvis ganvixiloT ori seqtori DCO ' da CND
pirveli seqtoridan ( DCO ' ) rLS ⋅Δ
=ρ
, (5.3.2)
6600
DOO 'Δ -dan ki βcosRr = ,
anu
βρ
cosRLS ⋅Δ
= . (5.3.3)
meore seqtoridan (CND )
βργ
ργ RctgDNS =⋅= , (5.3.4)
aqedan βργ tgRS
⋅⋅= ,
xolo (1) da (3)-dan miviRebT
βγ sin⋅Δ= L . (5.3.5)
rodesac geodeziuri grZedebis sxvaoba cnobilia vsargeblobT
(5.3.5) formuliT, xolo roca rkalis sigrZea cnobili, sargebloben (5.3.4)
formuliT.
Tu 200100 ÷=S m, maSin meridianTa Seaxloebas mxedvelobaSi ar iReben.
5.4. rumbebi
a) WeSmariti rumbi
rogorc aRvniSneT orientirebisaTvis gamoiyeneba WeSmariti azimuti
da WeSmariti rumbi. WeSmarit rumbs vuwodebT im umcires mosazRvre kuTxes,
romelsac mimarTuli qvedebuli adgens misi gamosavali wertilis WeSmarit
meridianTan.
nax. 35
N(ord) M1M4 NW : r4
NO : r1
O(st) W(est) O
SO : r2
SW : r3
S(ud) M2M3
6611
naxazze mimarTuli qvedebuli OM1-is umciresi mosazRvre kuTxe iqneba
r1. igi icvleba ο0 -dan ο90 -mde.
OM1 – NO . . .r1
OM2 – SO . . .r2
OM3 – SW . . .r3
OM4 – NW . . .r4
(indeqsi niSnavs kvadrantebs, r ki kuTxis odenobas)
b) damokidebuleba WeSmarit azimutebsa
da WeSmarit rumbebs Soris
vTqvaT mocemulia OM1 A1 azimuti, maSin rumbi iqneba NO:r (nax.43)
OM1 . . . A1 NO:r1 r1=A1
OM2 . . . A2 SO:r2 r2= ο180 _ A2
OM3 . . . A3 SW:r3 r3=A3 _ ο180
OM4. . . A4 NW:r4 r4= ο360 _ A4
aseTia damokidebuleba WeSmarit rumbebsa da WeSmarit azimutebs
Soris.
g) direqciuli rumbi
direqciuli rumbi aris umciresi kuTxe, romelsac mimarTuleba
adgens mis sawyis wertilSi gatarebul RerZa meridianis paraleluri
RerZis mimarT
nax. 36
M3 M2
O
r1
M1
y
r2
r3
r4M4
x'
6622
rodesac 'x RerZi daemTxveva RerZa meridians, gveqneba WeSmariti
rumbi, xolo roca daemTxveva RerZa meridianis paralelurs, maSin
gveqneba direqciuli rumbi.
d) damokidebuleba direqciul kuTxeebsa da
direqciul rumbebs Soris
OM1 . . . 1α . . . r1 r1= 1α
OM2 . . . 2α . . . r2 r2= ο180 _ 2α
OM2 . . . 3α . . . r3 r3= 3α _ ο180
OM4. . . 4α . . .r4 r4= ο360 _ 4α
Tu davakvirdebiT SevamCnevT, rom pirdapiri da Sebrunebuli rumbebi
icvlebian simetriulad e.i. kuTxuri sidideebi ucvlelia, xolo
kvadrantebi icvlebian simetriulad.
5.5. magnituri orientireba
magnituri orientirebis SemTxvevaSi qvedebulebi ukavSirdebian maTi
sawyisi wertilebis magnitur meridianebs, romlis ganxorcielebasac
magnituri isris magnituri RerZi iZleva (magnitur isars aqvs ori RerZi
geometriuli da magnituri).
cnobilia, rom dedamiwis sxvadasxva wertilebSi magnituri meridiani
ar emTxveva WeSmarit meridianebs. maT Soris iqmneba δ kuTxe, romelsac
uwodeben isris mixrilobas.
nax. 37
+δn N
S s
6633
magnituri azimuti aris kuTxe, romelsac mimarTeba qmnis magnituri
isris Crdilo mimarTulebasTan. igi icvleba ο0 -dan ο360 -mde saaTis isris
moZraobis Sesabamisad (magnituri isris polusebi aRiniSneba n da s-iT).
magnituri mixriloba aris kuTxe Seqmnili magnitur isarsa da
WeSmarit meridians Soris.
rodesac mixriloba aRmosavluria, igi aRiniSneba “+”-iT
A A1 = +m' 'δ . (5.5.1)
axla aviRoT dasavluri mixriloba. igi aRiniSneba “_”-iT
nax. 38
A A2 = −m'' ''δ (5.5.2)
WeSmariti azimuti udris magnituri azimutisa da mixrilobis
algebrul jams
A A A= ± = +m mδ δ . (5.5.3)
a) kavSiri direqciul kuTxesa da
magnitur azimutebs Soris
kavSiri direqciul kuTxesa da magnitur azimutebs Soris igivea, rac
kavSiri WeSmarit orientirebasa da magnitur orientirebas Soris. am
kavSiris dasadgenad viyenebT formulebs
6644
A = +α γ (5.5.4)
da
A A= +m δ . (5.5.5)
velze Cven vzomavT magnitur azimutebs, magram gegmaze viyenebT
direqciul kuTxes. zemoxsenebuli formulebidan
α γ δ+ = +Am , (5.5.6)
α δ γ δ= + − = −A Am m( ) ο , (5.5.7)
sadac δο aris magnituri mixrilobisa da meridianTa Seaxloebis sxvaoba,
anu magnituri isris mixriloba RerZa meridianis paraleluri RerZis
Crdilo mimarTulebasTan.
vTqvaT cnobilia Am , SegviZlia gamovTvaloT α an piriqiT.
δο-is gansazRvra siZneles ar warmoadgens.
ganvixiloT ramdenime SemTxveva
nax. 39
meridianTa Seaxloeba - +γ (aRmosavluria)
mixrilobac - +δ (agreTve aRmosavluria)
α δ γ δ= + − = +A Am m( ) ο, (5.5.8)
roca δ γ> .
6655
nax. 40
α δ γ δ= + − − − = −A Am m[ ( )] ο, (5.5.9)
roca δ γ< .
zogadad
α δ= +A m ο . (5.5.10)
e.i. direqciuli kuTxe udris magnituri azimutis da RerZa meridianis
paraleluri RerZidan magnituri isris mixrilobis algebrul jams.
5.6. cnobili azimutebiTa da direqciuli kuTxeebiT
mimarTulebaTa Soris kuTxeebis gansazRvra
nax. 41
6666
Tu cnobilia A1 da A2 azimutebi, maSin mimarTulebebs Soris kuTxe
β gamoiTvleba formuliT
β = −A A2 1 . (5.6.1)
Tu cnobilia ara azimutebi, aramed direqciuli kuTxeebi maSin
β α α= −2 1 . (5.6.2)
5.7. cnobili direqciuli rumbebiT astrolabiuri
kuTxeebis gansazRvra
astrolabiuri kuTxeebis gansazRvrisaTvis gamoviyenoT direqciuli
rumbebi.
varCevT oTx SemTxvevas
I – mocemulia r r1 4∩ da r r2 3∩
nax. 42
aq β1 1 4= +r r
β2 2 3= +r r
r3 r2
r4
β 2
r1
β 1
CW O
x
S
6677
II – mocemulia r r1 1∩ ' III – mocemulia r r1 2∩ da r r3 4∩
nax. 43 nax. 44
aq β = −r r1 1' β1 1 2180= − +ο ( )r r
β2 3 4180= − +ο ( )r r
IV – mocemulia r r1 3∩ da 2 4r r∩
nax. 45
aq 3 1
1 2 4
180 ( )
180 ( )
r r
r r
β
β
= − −
= − −
o
o (5.7.1)
astrolabiuri kuTxe ewodeba mimarTulebaTa Soris 180ο-ze nakleb
kuTxes.
maSasadame, rodesac rumbi simetriul kvadratebSia, maSin
mimarTulebaTa Soris astrolabiuri kuTxe udris 180ο minus udidesi da
umciresi kuTxeebis, anu rumbebis sxvaobas.
β
r1
r1'
O W
S
C
X
Oβ 2
r3
r1
r2
r4
β 1W C
S
X
6688
moyvanili 4 SemTxveva vrceldeba geografiuli da magnituri
rumbebis gamoyenebis drosac.
5.8. mixrilobis gansazRvra
mixrilobis gansazRvris ramodenime meTodi arsebobs. Cven
ganvixiloT yvelaze martivi. qaRaldze xazaven koncentrul wrexazebs,
romelTa centrSi amagreben 1-2 dm sigrZis Cxirs (an qinZisTavs) da mTeli
dRis ganmavlobaSi Tvals adevneben Cxiris Crdilis moZraobas.
nax. 46
aRniSnaven wrexazze Crdilis bolo wertilTa gadakveTas, cxadia
yvelaze grZeli Crdili iqneba mzis amosvlisas da Casvlisas (a da b) da
umoklesi ki SuadRisas (O). Tu SevaerTebT erTsa da imave wrexazze mdebare
wertilebs da monakveTebis Sua wertilebs xaziT; es xazi iqneba saZiebeli
meridiani. Semdeg yiblaniT vpoulobT magnituri meridianis mimarTulebas
ns da kuTxe am or mimarTulebas Soris aris saZiebeli mixriloba δ .
im wertilebis SemaerTebel xazs, romlebsac nulis toli
mixrilobebi aqvT, nulovani wertilebis SemaerTebeli xazi anu agonuri xazi
ewodeba. erTnairi mixrilobis wertilebis SemaerTebel xazs ki izogonuri
xazi ewodeba.
a b
c d
fe
Ss
W OO
Nn
σ
6699
VI Tavi
koordinatebi sainJinro geodeziaSi
koordinatebi ewodeba im saxeldebul fardobiT ricxvebs, romlis
saSualebiTac ganisazRvreba wertilebis mdebareoba.
garda cnobili koordinatebisa, romlebic Cven viciT (ciuri
koordinatebi), geodeziaSi ixmareba agreTve marTkuTxa, polaruli,
bipolaruli da gausis brtyeli marTkuTxa koordinatebi.
6.1. dekartis marTkuTxa koordinatebi RerZze
RerZs vuwodebT mimarTul xazs, romelzedac datanilia zomis
erTeulis Sesabamisi kvesurebi.
im SemTxvevaSi, roca RerZze daniSnulia gamosavali nulovani
wertili da masTan SedarebiT niSnebi – ricxvTa RerZs vuwodebT da
amboben, rom SemovitaneT dekartis sistema koordinatebisa
nax. 47
maTematikaSi miRebuli pirobis Tanaxmad nulidan marjvniv
miRebulia + niSani, marcxniv ki _ niSani.
RerZze yovel wertils Seesabameba erTi koordinati da yovel
koordinats erTi wertili, magaliTad:
A B( ), ( )+ −4 3
6.2. marTkuTxa koordinatebi sibrtyeze
a) marcxena sistema
sibrtyeze gvaqvs ori RerZi: abscisTa RerZi da ordinatTa RerZi.
winaswari SeTanxmebiT, maTematikaSi miRebulia, rom kvadrantebi
icvleba x RerZis mimarTulebidan y-is mimarTulebisaken mcire kuTxuri
manZiliT.
BB((--33)) 00 ricxvTa RerZi
AA((++44))
7700
nax. 48
yovel wertils sibrtyeze Seesabameba ori koordinati
A x y1 1 1( , )+ +
A x y2 2 2( , )− +
A x y3 3 3( , )− −
A x y4 4 4( , )+ −
b) marjvena sistema
geodeziaSi marTkuTxa koordinatTa sistema Semdegnairad
gamoiyeneba: radgan geodeziaSi orientirebisaTvis gamosavali RerZia
meridianis mimarTuleba, xolo maTematikaSi abscisTa x RerZi, amitom x
RerZis mimarTuleba daamTxvies meridianis mimarTulebas, rom maTematikaSi
miRebuli wesi ar darRveuliyo. aqac kvadrantebi icvleba x RerZis
mimarTulebidan, y-is mimarTulebisaken da koordinatebSi araviTari
gansxvaveba ar gveqneba, mxolod viRebT marjvena sistemas.
nax. 49
A1(+x1 +y1))
A2(-x2 +y2))
A4(+x4 –y4)
A3(-x3 –y3)
I
II
IV
III
x
y
A1(+x1 +y1)
A4(+x4 –y4)
A2(-x2 +y2)
A3(-x3 –y3)
I
IV
II
III
y
x
7711
A x y1 1 1( , )+ +
A x y2 2 2( , )− +
A x y3 3 3( , )− −
A x y4 4 4( , )+ −
6.3. marTkuTxa koordinatebi sivrceSi
dekartma sivrceSi gamoiyena sami RerZi: abscisebis, ordinatebis
da aplikatebis RerZebi.
nax. 50
vTqvaT, gvinda gavigoT A wertilis koordinatebi. gavatarebT
aRebul A wertilSi sam urTierTmarTob sibrtyes, ris Sedegad miviRebT x, y,
z koordinatebs, romelTac aqvT gadaadgilebis Tvisebebi
xyz
xzy
yxz
yzx
zxy
zyx
AA ((xx,,yy,,zz))
zz xx
yy
7722
cxrili (6.3.1)
oqtantebi x y z
I
II
III
IV
V
VI
VI
VIII
+
_
_
+
+
_
_
+
+
+
_
_
+
+
_
_
+
+
+
+
_
_
_
_
6.4. marTkuTxa koordinatebi sferoze
(zoldneris koordinatebi)
aviRoT sferoidi QQ1 ekvatoriT da PP1 RerZa meridianiT. marTkuTxa
koordinatebisaTvis nulovani wertili aq iqneba ekvatorisa da RerZa
meridianis gadakveTis F0 wertili.
nax. 51
D wertilis sferuli koordinatebia umoklesi (geodeziuri xazebi) xD
da yD xazebi.
garda amisa, D wertilis gansazRvrisaTvis unda vicodeT agreTve,
gamosavali F0 wertilis grZedi da ganedi F0 (B0L0). (nulovani imitom, rom
YYDD
XD Q
FF00
DD Q1
P1
P
7733
}
gamosavali wertili ekvatorzea. gamosavlad SeiZleba aRebul iqnas sxva
wertilic, oRond is iqneboda ara ekvatorze, aramed paralelze).
6.5. polaruli koordinatebi sibrtyeze
a) maTematikaSi
nax. 52
am sistemaSi sibrtyeze koordinatebi iqneba – ϕ polaruli kuTxe
aTvlili x RerZidan da λ radius veqtori.
sibrtyeze marTkuTxa koordinatebsa da polaruli koordinatebs
Soris damakavSirebeli formulebia:
x = λcosϕ
kvadrantebSi maTi niSnebi moyvanilia cxrilSi (6.5.1).
y = λsinϕ
cxrili (6.5.1)
kvadrantebi x
cosϕ y
sinϕ
I
II
III
IV
+
-
-
+
+
+
-
-
b) polaruli koordinatebi sibrtyeze
1 1 1( )A ϕl
4 4 4( )A ϕl A3 3 3( )λ ϕ
A2 2 2( )λ ϕ
λ1
ϕ 1
y
II
IIVVIIIIII
IIII x
2l
2ϕ
3l
4ϕ
4l
3ϕ
7744
(geodeziaSi)
nax. 53
aq polaruli ϕ kuTxis nacvlad gveqneba direqciuli kuTxe α ,
radgan x RerZi emTxveva RerZa meridians. amrigad, A wertilis
koordinatebi iqneba α da λ.
A( , )α λ
xolo niSnebi iqneba igive.
A1 1 1( , )α λ
A 2 2 2( , )α λ
A 3 3 3( , )α λ
A 4 4 4( , )α λ
g) polaruli koordinatebi sivrceSi
nax. 54
α _ aris direqciuli kuTxe
SS
A1 1 1( )α λ
A2 2 2( )α λA3 3 3( )α λ
A4 4 4( )α λ
λ1α 1
xx
II
IIII IIIIII
IIVV
yy OO WW
2l
4l
3l 2α
3α
4α
AA
zz
xx
yy
zz
αν
λ
oo
7755
ν _ daxris kuTxe
λ _ radius veqtori
e. i. A( , , )α ν λ
orientirebisaTvis saWiroa α direqciuli kuTxis gansazRvra;
mimarTulebis gansazRvrisaTvis saWiroa α da ν
mdebareobis gansazRvrisaTvis ki saWiroa α ν, da λ . aq sivrce iyofa
8 nawilad (oqtantebad).
naxazi 62-is mixedviT polaruli koordinatebis saSualebiT
marTkuTxa koordinatebis gamosaTvleli formulebia:
cos coscos sin
sin
xyz tg
ν αν α
ν ν
= ⋅= ⋅= ⋅ =
lll l
(6.5.1)
koordinatTa niSnebi kvadrantebisa da oqtantebis mixedviT
mocemulia (Sesabamisad) wina paragrafebSi mocemul (cx. 6.3.1 da cx. 6.5.1)
cxrilebSi.
d) polaruli koordinatebi sferoidze
nax. 55
aviRoT O wertili RerZa meridianze da mimarTuleba (romelic
warmoadgens geodeziur umokles xazs) OD (vertikaluri Wrili).
e.i. D wertilis mdebareobis gansazRvrisaTvis saWiroa: A – azimuti
da OD manZili (didi wris rkali).
D(A, OD)
orive am koordinats ewodeba D wertilis sferoiduli polaruli
koordinatebi.
P1
P
DO
E1 E
A
7766
6.6. bipolaruli koordinatebi
bipolaruli koordinatebis gansazRvrisaTvis gvaqvs ori SemTxveva:
1. kuTxuri gadakveTis da 2. xazovani gadakveTis
kuTxuri gadakveTa
nax. 56 nax. 57
xazovani gadakveTa
nax. 58 nax. 59
pirvel SemTxvvaSi gamoiyeneba O1 da O2 wertilebze gazomili β 1
da β 2 kuTxe an α 1 da α 2 direqciuli kuTxe, meore SemTxvevaSi ki λ1 da
λ2 radius veqtorebi.
esaa bipolaruli koordinatebi sibrtyeze. sivrceSi ki daemateba
mesame koordinati niSnulis (simaRlis) saxiT.
VII Tavi
pirobiTi aRniSvnebi
O O 1 1 OO22
A( ) λ λ 1 2
λ 2
λ 1
OO11 OO22
A H ( ) λ λ 1 2
λ 2
λ1
O O 1 1 OO22
A ( ) β β 1 2
β 2 β 1
O 1 O2
A ( ) α α 1 2
α 1
α 2
RRee rrZZ aa
mm eerrii ddii aa nnii ss ppaa rraa llee lluurrii
7777
7.1. gegmis Sedgena
vinaidan gegmis Sedgenis dros SeuZlebelia wvliladebis
gamoxazva Tavisi namdvili saxiT da odenobiT, amitom mimarTaven qaRaldze
maTi gamosaxvis dros pirobiTi aRniSvnebis gamoyenebas.
1. isini unda iyvnen msgavsi im wvliladebisa, rasac isini gamosaxaven.
2. unda iyvnen advilad gamosaxazavi.
3. ar unda iyos maTi raodenoba bevri da arc ise mcire, rom sagnebi
erTmaneTSi agverios.
mag., aviRoT tye, romelSic auarebeli jiSia. yvela jiSs Tavisi
pirobiTi aRniSvna rom mivuCinoT, maT rukaze ver gamovxazavT, rogori
msxvili masStabic ar unda iyos; Sesadgeni ruka an pirobiT tye moicavs
sxvadasxva masivebs: wiwviani, foTlovani, gadamwvari tye, buCqnari da sxva.
Cven, rom es masivebi erTi pirobiTi aRniSvniT gamovsaxoT, amiT masivebs
avurevT erTmaneTSi. Cveni saxalxo meurneobisaTvis ki aucilebelia
vicodeT, sad romeli masivi imyofeba.
pirobiTi aRniSvnebi gvxvdeba Semdegi saxis.
I fargiani (konturuli) pirobiTi aRniSvna, romelic pasuxobs kiTxvebs.
1. ra sagania?
2. sad mdebareobs?
3. rogori farTobi uWiravs?
4. rogori konfiguracia anu moyvaniloba aqvs? (mag. saxli, tba)
(nax.73)
nax. 60
II – saxis pirobiTi aRniSvnebi: ufargo (ara konturuli) upasuxebs
kiTxvebze:
1) ra sagania?
I gegma
1 : 1000
7788
sad aris?
(mag. gzis maCveneblis datana gegmaze romelic zustad
gviCvenebs ra sagania da sad aris)
III – saxis pirobiTi aRniSvna aris farTobiTi.
esaa gaerTianeba ufargo da farTobis (mag. fiWvis tyis an
venaxis masivSi dakavebuli teritoria)
IV – zoluri gvaZlevs saSualebas gavigoT
1. ra sagania?
2. sadaa?
3. ra sigrZe uWiravs?
(magaliTad, gzis sigrZe dadgindeba masStabSi zustad).
sagnebis (wvliladebis) ama Tu im jgufSi Setana damokidebulia
masStabze da saWiroebaze.
aviRoT zemoT moyvanili gegma, sxva 1
5000masStabSi
nax. 61
saxli, rom SevamciroT am masStabSi 5000 jer igi ar gamoCndeba, magram
gegmaze misi gamoxazva saWiroa da amitom saxls SevitanT meore jgufis
pirobiTi aRniSvnebiT. e.i. fargiani pirobiTi niSania, xolo meore gegmaze
15000
masStabSi ufargo pirobiTi niSania (II kategoria). an vTqvaT gzis
maCveneblis diametria 25 sm. I gegmaze masStabSi gamoCndeba, magram igi
gegmaze gamoixazeba ufargo pirobiTi aRniSvniT, radganac svetis sisqis
codna saWiro ar aris.
ra gansxvavebaa I gegmaze da II gegmaze fardobiT ufargo pirobiT
niSnebSi?
II gegma
1 : 5000
7799
is gansxvavebaa, rom II-ze aRebulia ufargo pirobiTi niSnebi mcire
zomis anu wvrili masStabisaTvis, xolo I-ze aRebulia, igive ufargo
pirobiTi niSnebi msxvili masStabisaTvis.
uswormasworobis anu reliefis gamomsaxveli pirobiTi aRniSvnebi
unda akmayofilebdnen Semdeg pirobebs:
1. isini unda gvaZlevdnen saSualebas gavigoT gegmaze yoveli
wertilis simaRle (niSnuli);
2. unda gvaZlevdes saSualebas gavigoT daqanebis mimarTuleba;
3. rogoria daqanebis xarisxi;
4. unda gvaZlevdes warmodgenas zedapiris plastiurobis Sesaxeb
(sadaa Sezneqili, gamozneqili).
uswormasworobis gamomsaxveli pirobiTi aRniSvnebi aseTia:
1. niSnulebi,
2. izohifsebi,
3. niSnulebi da izohifsebi erTad,
4. kvesurebi,
5. punqtirebi,
6. SeRebva (Seferva).
viciT, rom niSnuli aris ricxvi, romelic gamoxatavs wertilis
simaRles donebrivi zedapiridan.
aviRoT A da B wertilebis HA da HB-absoluturi (normaluri)
simaRleebi (nax. 62).
gavataroT pirobiTi donebrivi zedapiri 'A wertilisaTvis. H1 – H2-
pirobiTi simaRleebi iqneba.
nax. 62
8800
vTqvaT donebrivi zedapiri gavataroT A wertilSi. h-iqneba
aRmateba. anu fardobiTi simaRle (igi SeiZleba iyos rogorc + niSniT, ise –
niSniT) maSasadame niSnulebi gvxvdebian:
1. absoluturi,
2. pirobiTi,
3. fardobiTi (aRmateba).
viciT, rom topografiuli zedapiris funqcia gamoisaxeba ase
),( yxZ ϕ=
da xasiaTdeba calsaxobiT, sasrulobiT, uwyvetobiT da
mdovrulobiT.
7.2. niSnulebiani gegmilebi
a) wertili sivrceSi
nax. 63
vTqvaT, gegmaze gvaqvs sami wertili a5, b8, c4. Sesabamisad 5, 8, (-4)
niSnulebia (koordinatebia), rac gvaZlevs warmodgenas reliefze. Z – mesame
koordinatia. viciT, rom sivrceSi wertilis mdebareobis gansazRvrisaTvis
saWiroa 3 koordinati. x, y, koordinats ki ase warmovidgenT. gegmaze
pirobiT an fiuzos Sesabamisad avagebT koordinatTa RerZebs da yvela
wertilis mdebareobas vsazRvravT am RerZebis mimarT (nax. 63) e.i. x da y-s
vTvliT cnobilad, xolo mesame koordinati iqneba niSnuli 5, 8, -4, romlebic
gegmis zemoT da qvemoT gadaizomeba masStabSi.
8811
b) xazi sivrceSi
xazebs sivrceSi SeiZleba eWiros nebismieri mdebareoba. isini
SeiZleba iyvnen: paraleluri, acdenili, gadakveTili, urTierTmarTobi.
magaliTad, gegmaze gadakveTili xazebi sivrceSi SeiZleba iyvnen
acdenilni, gadakveTilni, aseve gegmaze paraleluri xazebi sivrceSi
SeiZleba iyvnen paralelurebi an acdenilni. an kidev gegmaze Serwymuli
xazebi, sivrceSi SeiZleba iyvnen paraleluri, gadakveTili da marTobi.
ganvixiloT isini cal-calke.
aq ZiriTadad gvxvdeba ori SemTxveva:
1. mocemulia a5, b8 xazis ori wertilis sam-sami koordinati A(xA, yA, zA)
da B(xB, yB, zB)
2. meore SemTxvevis dros sivrceSi xazis mimarTuleba ganisazRvreba
erTi (a5) wertilis sami koordinatiT A(xA, yA, zA) da mimarTulebebiT.
nax. 64
A0
B0
a5
b8
X
δ
a
h L
L
L
B(xb yb zb)
A(xa ya za)
aa55
α δαα
∩∪ ∩∪ ∩
iL
bb88
8822
α δ∩ (mimarTebis kuTxe, daxris kuTxe)
Υ
α δ∩ (mimarTebis kuTxe, daxris kuTxe)
Υ
α ∩ i (mimarTebis kuTxe, qanobi)
Υ
α ∩ L (mimarTebis kuTxe, intervali)
α - kuTxe SeiZleba iyos nebismieri saorientacio kuTxe: azimuti,
rumbi, direqciuli kuTxe da sxva. α - mimarTebis kuTxe aris Semdgari
xazis mimarTebis mier x RerZis dadebiT mimarTulebasTan 0ο -dan 360ο -mde.
xazis dadebiT mimarTulebad ki pirobiTad iTvleba mimarTuleba
daRmarTisaken (e.i. b8-dan a5-ken). samTo (geometriaSi) saqmeSi δ - aris
daxris kuTxe (CvenTvis cnobilia ν ) mimarTuli monakveTis mier Seqmnili
mis qvedebulTan.
i -aris aRmateba - h-is fardoba qvedebulTan
i ha L
= =1 (7.2.1)
i -aris tgδ ,
L -s ewodeba intervali.
intervali aris qvedebuli, romlis Sesabamisi aRmateba=1. maSasadame
sivrceSi xazis mimarTebis gansazRvrisaTvis saWiroa ori wertilis
koordinatebi an erTi wertilis koordinatebi da mimarTeba (α an δ )
A0
B0
δ
B0
A0 δ
8833
nax. 66 nax. 67
am naxazebSi gansxvavebaa x da y-Si. α (direqciuli anu mimarTebis)
kuTxe cvlis or x da y koordinats, xolo δ cvlis z-s
nax. 68
δ -s vigebT transportiriT
i -s vigebT gazomviT da gamoTvliT
L -s ki graduirebiT
vTqvaT mocemulia xazebi 5-8, 8-14, 6-12 da unda gavigoT δ kuTxe.
amisaTvis avagoT xazebi. kveTis simaRle h=1m.
xazi, romlis L-ic didia, is naklebad daqanebulia. romlis
intervalic mcirea, is ufro daqanebulia.
L1
L1
L2
L2
L2
L2
L2 L3
L3
L3
L3
L3
δ δ δ
h
h
i =3
1α
2
2
6ia
= 33
6ia
=
a) b) g) h
26
24
L2 L3
δ 2 22
30
L1
δ 1
δ 3
8844
nax. 69 (a, b, g)
g) graduireba
graduirebisaTvis varCevT or SemTxvevas.
1. roca mocemulia erTi wertilis koordinatebi, xazis mimarTeba da
kveTis simaRle.
2. roca mocemulia ori wertilis koordinatebi (niSnulebi) da
kveTis simaRle.
I SemTxveva. vTqvaT mocemulia wertili (75,6); α = 135ο ; δ = 25ο , e.i.
wertilis koordinati da mimarTuleba α da δ
gavataroT Tarazuli xazi da
transportiriT avagoT 25ο kuTxe masStabis Sesabamisi sigrZe. aviRoT 1m da
Sesabamisi L intervali davyoT 10 nawilad. L-is 0.6 gadavzomoT
daRmarTisaken, miviRebT 75 niSnulis mqone wertils, Semdeg gadavzomoT 75-
dan mTliani L-is toli miviRebT 74-s; xolo 75.6-dan gadavzomoT
aRmarTisaken L-is 0.4-di, miviRebT 76 wertils. graduirebis mizania ara
L
L
0,4
0,6
77
76
75,6
75
74
73
ο135=a
L
1mδ = 25ο
nax. 70
8855
1111,,44
55,,22
66 77
88
1111
1100
99
nax. 71
1111,,88
1111,,22
1111,,88 1111,,22 00,,66
nax. 73
nnaaxx.. 8822
11,4
10,8
11,4 10,8 0,6
marto is, rom SevadaroT intervalebi, aramed erTnairi niSnulis mqone
wertilebi SevaerToT da gamovsaxoT reliefi.
graduireba niSnavs davniSnoT xazze iseTi wertilebi, romlebic
jeradi iqneba kveTis simaRlisa.
II-SemTxvevaSi rodesac mocemulia xazis ori wertilis sam-sami
koordinati. am SemTxvevaSi x, y-s vTvliT cnobilad da z koordinatiT
vaxdenT graduirebas aq varCevT sam xerxs.
1. uSualo interpoloba
2. trafaretis anu serseroTis xerxi
3. profilis anu ujrediani qaRaldis
xerxi
1) vTqvaT mocemulia ori wertili 11.4 da
5.7;
sxvaoba 11,4-5,2=6.2≈ 6 nawilad gavyofT mocemul xazs da 6.2 wertilidan
gadavzomavT ⎟⎠⎞
⎜⎝⎛
61
monakveTis 0.8-s, miviRebT 6
wertils. 11.4-dan dadablebisaken gadavzomavT
0.4-s miviRebT 11 wertils. Semdeg 11 da 6
wertilebs Soris manZils vyofT 11-6=5
nawilad da wavawerT Sesabamisad 7, 8, 9, 10
niSnulebs. rodesac wertilis niSnulebs
Soris sxvaoba naklebia kveTis simaRleze,
magram am wertilebs Soris aris kveTis
simaRlis jeradi niSnulis Sesabamisi
wertili. maSin damrgvalebas ar vaxdenT da vawarmoebT 6 aTwilad
nawilebze mocemuli xazis dayofas (nax. 72).
rodesac wertilis niSnulebs Soris sxvaoba naklebia kveTis
simaRleze da am wertilebs Soris ar aris kveTis simaRlis jeradi
niSnuli, maSin xazis graduirebas ar
vaxdenT. (nax. 73)
2) trafaretis anu serseroTis
xerxi damyarebulia im geometriul
nax. 72
8866
5,7
12
11
10
9
8
7
6
5
11,4
11,4
5,7
11 22
11 11
11 00
99
88
77
66
55
11 11 ,, 44 11 11 ,, 44
55 ,, 77
11 22
11 11
11 00
99
88
77
66
55
11 11 ,, 44 11 11 ,, 44
55 ,, 77
11,4
mosazrebaze, rom Tu monakveTs gadavkveTT Tanabrad dacilebuli
paraleluri xazebiT, maSin (monakveTi) xazi daiyofa tol monakveTebad.
serseroTis xerxisaTvis winaswar vamzadebT gasanTlul qaRaldze
paralelur xazebs toli manZiliT dacilebuls. ixileT (nax. 74 a, b, g)
a) b) g)
nax. 74 (a, b, g)
vTqvaT gvinda vawarmooT interpoloba 5.7 da 11.4 wertilebs
Soris. amisaTvis winaswar serseroTze wavawerT saWiro niSnulebs 5, 6, 7, 8, 9,
10, 11, 12. 5-sa da 6-s paralelur xazebs Soris TvaliT SevafasebT da
davniSnavT 5,7 wertils. 11-sa da 12-s Soris gavavlebT (wyvetiliT) 11.4
Sesabamis xazs. Semdeg xazis 5,7 davamTxvevT serseroTze aRniSnul 5,7
wertils da serseroTs vabrunebT 11.4-is gadakveTamde mere 6, 7, 8, 9, 10, 12
niSnulebiT Sesabamis xazebsa da xazis gadakveTis wertilebs davaWerT
fanqars da xazs movaSorebT serseroTs da bolos fanqriT naWdevze
wavawerT 6, 7, 8, 9, 10, 11. es meTodi interpolobisa aris sakmaod zusti.
serseroTebi mzaddeba sxvadasxva zomis. 10 mm-iani, 5 mm-iani da sxva.
Cvens SemTxvevaSi I-li zomis serseroTi ar gamodgeba, radgan 5,7
wertilSi serseroTis brunviT 11.4-is xazs ar gadakveTs, radganac is
moklea.
saerTod, xazisa da serseroTis gadakveTa rac ufro marTobulad
iqneba, miT metia sizuste.
e.i. Cveni SemTxvevisaTvis III-e (g varianti) ivargebs, magram naklebi
sizustis iqneba.
8877
d) profilis anu ujrediani – (milimetrebiani)
qaRaldis xerxi
nax. 75
mocemulia xazi (5,7–11,4). movaxdinoT misi graduireba
milimetrebiani qaRaldiT.
santimetrebian danayofebs wavaweroT niSnulebi 5, 6, 7, 8, 9, 10, 11, 12;
Semdeg milimetrul qaRaldze mivadoT gverdidan dasagraduirebeli xazis.
5,7 wertili da 11.4 avagegmiloT milimetrul qaRaldze. Semdeg 12.4 da 5.7
wertilebi SevaerToT xaziT. gadakveTis wertilebi 6, 7, 8, 9, 10, 11, 12
santimetrebian xazebidan CamovagegmiloT 5.7 – 11.4 – xazze, miviRebT
dagraduirebul xazs (gadakveTis wertilebze monakveTebis fargliT
gadazoma da ise gadatana dasagraduirebel xazze mizanSewonili araa, igi
Secdomebis wyaroa). graduirebis es xerxi kidev ufro zustia.
saerTod geodeziaSi (sizustis Sesabamisad) vmuSaobT aucileblobisa
da sakmarisobis principis Sesabamisad, amitom graduirebis xerxs
SevarCevT imis mixedviT, Tu rogori sizustiTaa saWiro samuSaos
Sesruleba. Tu wvril masStabSi vmuSaobT, viyenebT uSualo interpolobas,
saSualo masStabSi, serseroTs, xolo msxvil masStabSi milimetrebiani
qaRaldis xerxs.
12
5
6
7
8
9
10
11
5,7 6 7 8 9 10 11 11,4
5,7
11,4
8888
7.5. niSnulebiani proeqciebis ganzogadoeba
Tu topografiul zedapirs gavkveTT paraleluri horizontaluri
erTmaneTisagan toli manZilebiT daSorebuli sibrtyeebiT, maSin kveTaSi
miviRebT erTmaneTis paralelur mrud wirebs an wrfeebs – imisda mixedviT,
Tu rogoria topografiuli zedapiri. ganvixiloT magaliTebi:
1) 2) 3)
Tanabari daqanebis amozneqili cilin- Cazneqili cilin-
mqone zedapiri druli zedapiri druli zedapiri
4) 5) 6) 7)
nax. 76 nax. 77 nax. 78
nax. 79 nax. 80 nax. 81 nax. 82
8899
qedi. wyalgamyofi
dadebiTi invari-
antuli xazi Ta-
nabari daxriT
(izohifsebi Ta-
nabradaa dacile-
buli).
Rele. Tanabrad
daqanebuli wyal-
Semkrebi anu uar-
yofiTi invariant-
tuli xazi.
qedi. wyalgamyofi
anu dadebiTi inva-
riantuli xazi
ara Tanabari dax-
riT (izohifsebi
araTanabradaa
dacilebuli).
Rele. Tanabrad
daqanebuli wyal-
Semkrebi xazi anu
uaryofiTi invari-
antuli xazi ara
Tanabari daxriT.
nax. 83
gavarCioT zemoT gamosaxuli yvela SemTxveva cal-calke:
1-el magaliTSi mocemulia Tanabari daqanebis mqone mTis ferdobis
gankveTa (warmodgeniT) horizontuli sibrtyeebiT. kveTaSi miiReba swori –
erTmaneTisagan Tanabrad daSorebuli paraleluri xazebi (e.i. gegmaze
Tanabrad daSorebuli paraleluri xazebi rom gveqneba unda vicodeT ras
warmoadgens is adgilze).
2-e SemTxvevaSi zedapirs aqvs amozneqili cilindris forma. kveTaSi
miiReba erTmaneTisgan araTanabrad daSorebuli wrfeebi.
3-e SemTxvevaSi zedapiri Cazneqilia da izohifsebic araTanabrad
iqneba daSorebuli.
4-e SemTxvevaSi gvaqvs Tanabari daqanebis mqone qedi (sami mimarTulebiT
daRmarTi, erTi mimarTulebiT aRmarTi. izohifsebi TiTqos gagvirbian).
safexura anu terasa
(sadac erTmaneTs
enacvleba daqaneba da
wavakeba)
8)
9900
5-e SemTxvevaSi mocemulia Rele wyalSemkrebi anu uaryofiTi
invariantuli xazi. gvaqvs sami mimarTulebiT aRmarTi, erTi mimarTulebiT
daRmarTi.
6-e naxazze mocemulia ara Tanabari daqanebis mqone qedi, gankveTis
wirebi (izohifsebi) erTmaneTisagan ara Tanabradaa daqanebuli.
7-e naxazze gvaqvs ara Tanabari daqanebis Rele.
8-e naCvenebia safexura anu terasa.
nax. 84
zemoT moyvanili SemTxvevebi ganvazogadoT ufro rTul zedapirze.
vTqvaT, gvaqvs unagira, e.i. adgili erTdeba ori qediT da iwyeba ori Rele.
Tu gvaqvs konusuri zeda-
piri kveTaSi, dagegmile-
bisas miviRebT koncentrul
wrexazebs.
9911
nax. 86
nax. 85
rogorc vxedavT, izohifsebi arian Sekruli mrudebi toli simaRlis
mqone wertilebiT, romelTa niSnulebi jeradia kveTis.
Tu davakvirdebiT izohifsebs,
davinaxavT, rom (mTa) gora gamosaxulia
urTierT paraleluri izohifsebiT, magram
imave izohifsebiT SeiZleba gamosaxuli
iqnes tafobic. ismis kiTxva: rogor unda
gavarCioT gegmaze mocemuli izohifsebi –
ras gamoxataven tafobs Tu goras? aq
yuradReba unda mivaqcioT niSnulebs.
rodesac izohifsebze niSnulebi Suisaken
izrdeba, maSin izohifsebiT amaRlebuli
adgilia gamosaxuli, rodesac piriqiTaa,
maSin adgili tafobs warmoadgens. an
izohifsebze SeiZleba saerTod ar iyos niSnulebi. igi SeiZleba kvesurebiT
iyos aRniSnuli ) dadableba ) amaRleba, an ubralod izohifsebze
SuaSi eweros (+) an (_) niSani.
izohifsebi gvaZleven saSualebas nebismieri wertilis niSnulis
gansazRvrisaTvis.
Cven viciT, rom uswormasworobis gamomsaxveli pirobiTi niSnebi
unda gvaZlevdnen saSualebas ganvsazRvroT:
1. nebismieri wertilis niSnulebi;
9922
nax. 87
nax. 88
1. saiTkenaa daqaneba;
2. rogoria daqanebis xarisxi;
3. unda gvaZlevdes saSualebas war-
modgena gvqondes reliefis plas-
tikurobaze.
SeiZleba izohifsebi akmayofilebs
yvela am pirobas, magram mas mainc aqvs
nakli. igulisxmeba, rom or mezobel
izohifss Soris Tanabari daqanebaa, rac
swori araa (nax. 87), arc kveTis simaRlis
Semcireba gvaZlevs sasurvel Sedegs.
ganvixiloT kveTis simaRlis Semcirebis
magaliTi safexuraze (nax. 88)
safexura anu terasa gamoisaxeba
axlos ganlagebuli da mkveTrad daSorebuli izohifsebisagan.
ganvixiloT SemTxvevebi:
I – SevamciroT kveTis
simaRle, maSin damxmare izohifsebi
20-30 Soris ganawildeba, rogorc es
naxazzea gamosaxuli, magram,
rogorc zemoT ganxilulidan
viciT, Tanabrad daSorebuli
izohifsebiT igulisxmeba ara
iseTi daqaneba rogoric safexuras
gaaCnia, aramed Tanabari daqaneba AB
xazis gayolebiT. Tu davakvirdebiT
zedapirze arsebul 25-26 da 30
wertilebs davinaxavT, rom isini ar
Seesabamebian Cvens mier gegmaze
kveTis simaRlis SemcirebiT miRebul izohifsebs. axla mivaqcioT
yuradReba 20, 25, 26 da 30 wertilebs. interpoloba gegmaze gavakeToT
aRebuli wertilebis mixedviT, maSin gegma miiRebs I-is nacvlad II-re saxes
(nax. 88) romlebic arafriT ar emsgavsebian erTmaneTs. maSasadame
9933
mniSvneloba aqvs damaxasiaTebeli wertilebis niSnulebis gansazRvras
velze da ara kveTis simaRlis Semcirebas, mxolod amis Semdeg SegviZlia
vimsjeloT kveTis simaRlis SerCevaze.
a) normaluri kveTis simaRlis gansazRvra
45ο -aris zRvruli daqaneba, romlis
zeviT daqanebis SemTxvevaSi, zedapirs
izohifsebiT ver gamovsaxavT.
vTqvaT gvaqvs 45ο -iani daqaneba.
normuli kveTis simaRle damokidebulia
ferdobis zRvrul (kidegan) daqanebaze da
gegmis masStabze. zRvrul (anu kidegan)
daqanebad miRebulia daqaneba 45ο .
zRvruli siaxlove or izohifs Soris
aris 0.01 sm. kveTis simaRle zRvruli iqneba 0.01 sm anu roca masStabia
110000
, h = 1m. rodesac daxris kuTxe 45ο -is nacvlad aris 22 5ο , maSin kveTis
simaRle izohifsebs Soris imave dacilebisaTvis unda aviRoT h2 (orjer
naklebi).
rogor SevarCioT kveTis simaRle?
amisaTvis gavarCioT ori SemTxveva 22ο da 45ο -Tvis
I) – vTqvaT kveTis simaRlea – 1m.
45ο -ze rodesac h = 1m
izohifsebs Soris manZili 0.01 sm. e.i.
udris zRvrul grafikul sizustes.
imave SemTxvevisaTvis 22ο-ze manZili
izohifsebs Soris gaizrdeba e.i.
rodesac kveTis simaRle h = 1m. 45ο -ze
izohifsebs Soris manZili zRvrulia,
xolo 22ο-ze izohifsebs Soris
zRvrul dacilebas Seesabameba h2
kveTis simaRle.
nax. 90
0,01 sm
h=0,01 sm
22 ,5o
nax. 89
0,01 sm
0,01
45ο
9944
nax. 92
nax. 91
(45ο -ze h = 1m kveTis simaRlis dros izohifsebs Soris dacileba
zRvrulia, magram h2 kveTis simaRlis
dros 45ο -ze izohifsebs Soris manZili
0.01sm-ze naklebi gamodis, ris gamo
izohifsebi Seirwymian. maSasadame 45ο -
isaTvis rodesac h = 1m. dasaSvebia,
(magram h2 ara)
II) axla ganvixiloT 22 5ο, -ze
Tu h = 1m izohifsebs Soris manZili
0.02sm. e.i. gaorkecebul zRvrul
sizustes, rodesac h2, maSin izohifsebs
Soris manZili udris 0.01 e.i. kveTis
simaRle unda aviRoT 45ο -is Sesabamisi
da ara 22 5ο, -isa.
kveTis normaluri simaRle reliefisaTvis unda SeirCes.
saerTod Tu adgili
aqvs sxvadasxvagvar daqane-
bas, maSin kveTis simaRle
unda SevarCioT maqsima-
luri daqanebisaTvis da is
gamodgeba yvela danarCeni
daqanebebisaTvis. saerTod,
mTavaria pirvel rigSi
damaxasiaTebeli werti-
lebis SerCeva, xolo
Semdeg kveTis simaRlis
SerCeva.
davamyaroT damokide-
buleba a-qvedebulsa, ν
daxris kuTxesa da h kveTis simaRles Soris. a h ctg= ⋅ ν
h/2
45ο 22 5ο,
h
0.01
45ο 22 5ο,
I
II
9955
SevadginoT cxrili
ν
h 1ο 2ο 3ο 4ο 5ο 10ο 15ο 20ο 25ο 30ο 35ο 40ο 45ο
1m 57.3 28.6 19.1 14.3 11.4 5.7 3.7 2.7 2.1 1.7 1.4 1.2 1.0
qvedebulebi gamovTvaleT formuliT a h ctg= ⋅ =ν ν( , ... )1 2 45ο ο ο kuTxis
zrdisas manZili araproporciulad mcirdeba, xolo zogierT intervalebSi
daculia proporciulad Semcireba.
vTqvaT, gvinda gzis gayvana, romlis daxraa 7ο . gamovTvaloT
Sesabamisi a. ganvsazRvroT qvedebulis kleba 1ο-iT zrdisas 5ο-dan 10ο-mde.
1 114 575
114ο −−
=. . . m
axla ganvixiloT Sebrunebuli amocana. mocemulia a=9.12 da saWiroa
ganvsazRvroT daxris ν kuTxe.
11.4-9.12=2.28
2.28 : 1.14=2
e.i. ν = + =5 2 7ο ο ο
nax. 93
agebuli diagramis mixedviT ganvsazRvroT cxrilSi Seutaneli
daxris kuTxis Sesabamisi qvedebuli ν RerZze. gadavzomavT am kuTxis
Sesabamis manZils, miRebuli wertilidan aRvmarTavT marTobs diagramis
gadakveTamde da miviRebT qanobs mocemul masStabSi.
qvedebulis gansazRvra SeiZleba qanobis mixedviTac
ha
tg i= =ν a hi
= i = =0 01 1%.
0.001 1%i = = o.
gavutoloT h=1m da SevadginoT cxrili
aa
νν 10 20 30 40 50 100 150 200 250 300 350 400 450
9966
i
h 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012
1m 1000 500 333.3 250.0 200.0 166.7 142.8 125.0 111.1 110.0 90.9 83.6
nax. 94
cxrilis saSualebiT aigeba diagrama: cxrilSi winaswar mocemuli
h-is da qanobis sxvadasxva mniSvnelobebisaTvis gamoiTvlian qvedebulebs
formuliT
a hi
= roca hi
==
10 001m
.
a = =1 1000m
0.001m da a.S.
diagramis asagebad horizontul xazze nebismieri masStabiT
movzomavT qanobebs, xolo Sveuli mimarTulebiT ki mocemul masStabSi
qvedebulebs. miRebul wertilebs SevaerTebT, ris Sedegadac miviRebT
saZiebel diagramas.
VIII Tavi
aa,,mm
i %%o
9977
nax. 95
nax. 96
Tarazoebi
8.1. Tarazos Teoria
xazebisa da sibrtyeebis horizontalur da vertikalur mdgo-
mareobaSi mosayvanad sargebloben TarazoTi. geodeziuri instrumentebi
Sedgeba RerZebisa da wredebisagan, romelTa dayeneba Sveulad an
Tarazulad aucilebelia
samuSaoebis Sesrulebis
dros. am amocanis gadasaw-
yvetad gamoiyeneba Tara-
zo.
Tarazo ZiriTadad orgvaria: cilindruli da sferuli.
cilindruli Tarazoebic
ori saxisaa:
1. pirveli saxis
cilindruli Tarazo (xe-
losnebis Tarazo) ase mzad-
deba: aiReben SuSis mils,
gaaxureben da sasurveli
radiusiT moRunaven. masSi
Caasxamen spirts, toveben
cota haers buStulasaTvis
da daxSaven. milis simrudis
maqsimaluri radiusi R max ≈ 2
metramdea.
1-li ampuliT sibrtye moyavT Tarazulad, 2-Ti ki vertikalurad.
TvaliT interpolobisaTvis Tarazos zemoT aqvs ori Strixi.
2. meore saxis cilindruli Tarazo aris ufro zusti. mis
dasamzadeblad iReben mils, Sig moaTavseben garkveuli profilis mqone
qlibs, romlis simrudis radiusi aRwevs 200 metrs. am qlibs abruneben
milSi. milis Siga zedapiri miiRebs qlibis profils. Tu mili orive
mxareze aris gaxexili, maSin gveqneba ormxrivi anu reversiuli Tarazo.
Tarazos milakze zrdad warwerebs akeTeben Suidan marjvniv da marcxniv
an erTi mxriT. naxazze mocemulia reversiuli (ormxrivi) Tarazo orgvari
warweriT. miRebulia Tarazos ormilimetriani danayofebi.
9988
nax. 97
nax. 98
mils erTi mxridan daxSaven minis sagozaviT da Sig Caasxaven
gogirdovan eTers, ise rom gavsebas cota akldes, aTavseben 30 40ο ο− -mde
gacxelebul qviSian an marilian abazanaSi. mili siTxis gafarToebis
Sedegad aivseba da daiwyebs gadmosvlas, am dros mils daxSaven. gacivebis
Semdeg siTxe SeikumSeba da gaCndeba uhaero buStula. ase damzadebul
mils aTavseben sasurveli formis CarCoSi. CarCos ukeTeben Tarazos
Semasworebel xraxns. Tarazos aqvs 2 RerZi: 1. geometriuli anu simetriis
RerZi AB da 2. TviT Tarazos RerZi HH1, romelic aris nul punqtis mxebi
Signidan.
maSasadame, Tarazos HH1 RerZi ewodeba im wrfes, romelic
warmoadgens Siga mxebs nul
punqtSi. Tarazos gaaCnia 2
sakani Tarazos orive mxri-
saTvis (erTi sakani zeda
mxaresaTvis, meore qveda
mxaresaTvis), sakani gamoiye-
neba zamTarSi an zafxulSi
siTxis saregulireblad.
II sferuli Tarazo
sferuli Tarazo warmoadgens sferos nawils (naxazze mocemulia
WrilSi) Tarazos RerZia OC.
Tarazos aqvs Semasworebeli
xraxni.
Tarazos RerZis daxris
kuTxis gansazRvra
vTqvaT, Tarazos O (nul)
punqts emTxveva buStulas m
centri (aq laparakia orive
cilindrul da sferul Tara-
zoze), maSin naxazze HH cilin-
druli da OC sferuli Tarazos RerZebs eqnebaT Sesabamisad hori-
zontaluri da vertikaluri mdgomareobebi. OC sferuli Tarazos RerZi
9999
davxaroT θ kuTxiT, cilindruli Tarazos H H' ' RerZi ki α kuTxiT e.i. HH-
ma miiRo H H' ' mdebareoba. xolo OC-em miiRo O C' mdebareoba. rogorc
naxazidan sCans α θ= .
erTi 2 milimetriani danayofis Sesabamisi centraluri kuTxe
avRniSnoT τ -Ti, rasac Tarazos safasuri ewodeba. naxazidan θ α τ= = ⋅ n , e.i.
Tu viciT τ da nul punqtidan buStulas Suagulis gadaxris Sesabamis
danayofTa n ricxvi, miviRebT α anu θ kuTxes.
vinaidan nul punqtidan buStulas Suagulis Sesabamisi anaTvlis
aReba zustad Znelia, amitom vsargeblobT misi boloebis Sesabamisi
anaTvlebiT (buStulas marjvena da marcxena boloebiT), amasTan
dakavSirebiT ganvixiloT ori SemTxveva.
1. buStula aris nul punqtidan marcxniv
(naxazze Tarazos RerZis marjvena mxarea
daxrili), davweroT am SemTxvevisaTvis om
manZilis gamosaTvleli formula. naxazidan
om oc cm oc oc od oc od= − = −
−=
+2 2
(8.1.1)
2. ganvixiloT SemTxveva, roca
buStula OO-punqtidan marjvniv gadadis, e.i.
roca anaTvali aiReba O-punqtidan orive
mxares. maSin naxazidan.
2 2oc od oc odom oc cm oc + −
= − = − = (8.1.2)
manZilebi gamovsaxoT anaTvlebiT.
aRvniSnoT
oc - manZili a danayofis raodenobiT,
od - manZili b danayofis raodenobiT,
om n= .
danayofebi aTvlili O-punqtidan marcxniv aRvniSnoT “+”-iT, O-
punqtidan marjvniv ki “_“-iT, e.i. (1) da (2) gagrZeldeba ase:
n a b=
+2
(nax. 99) n a b a b=
− −=
+( )2 2
(nax. 100)
nax. 99
o
d c m
o d c
m
nax. 100
110000
nax. 101
nax. 102
danayofebis raodenoba O-punqtidan buStulis Suagulamde udris
buStulis boloebis Sesabamisi anaTvlebis algebrul saSualos.
aviRoT reversiuli Tara-
zos zedxedi orive mxridan da
ganvsazRvroT n-anaTvali.
I-dan
n a b=
+=
− + −= −
22 8
25( ).
II-dan
n k a b=
− −=
− −= −
22
20 12 182
5 ,
k -manZilia Skalis sawyisidan
O-punqtamde.
Tarazos danayofis safasu-
ris gansazRvra.
Tarazos safasuris
gansazRvrisaTvis iyeneben
egzamenators. garegnulad
igi reisSinas waagavs. A da
B wertilebSi xistad
Camagrebulia wvetanebiT. C
wertilSi gayrilia xraxni, romlis Tavi dayofilia m nawilad.
nax. 103
avRniSnoT egzamenatoris safasuri "Эτ -Ti, danayofebis raodenoba
ki, rogorc viciT, aris m. gamosacdel Tarazos aTavseben egzamenatorze,
rogorc naCvenebia 103 naxazze.
110011
nax. 104
h_aris xraxnis biji, romelic Seesabameba mis erT srul
Semobrunebas da ganisazRvreba Semdegnairad. aviRoT xraxni da davaWiroT
qaRaldze. igi datovebs kvals. gavzomoT H manZili kidura kvalebs Soris
da davTvaloT maT Soris ramdeni xraxnis bijia – N . maSin
h HN
= .
Sesabamisi centraluri kuTxe udris
"Эmτ -s. sanam Tarazos safasurs
ganvsazRvravdeT, manamde unda
ganisazRvros "Эτ naxazidan
"ρτλhm Э = ; da "" ρτ
mh
Э ⋅=
λ (8.1.3)
sadac λ-gaizomeba, m-danayofTa raodenobaa egzamenatoris xraxnze;
"206000"=ρ -radiania sekundebSi; h -biji gansazRvrulia.
Tarazos safasuris "τ gansazRvrisas viqceviT ase: egzamenatoris
xraxns vatrialebT manamde, sanam buStula mTlianad ar gadava nul
punqtidan marcxniv, aviRebT anaTvlebs a1 da b1 Tarazoze da m1
egzamenatorze, vatrialebT xraxns ise, rom Tarazos buStula gadavides
nulpunqtidan marjvniv mTlianad da aviRebT anaTvlebs a2 da b2 Tarazoze
da m2 egzamenatorze.
buStulas centris Sesabamisi anaTvali pirvel SemTxvevaSi iqneba
211
1ba
n+
= ; Sesabamisi anaTvali m1 egzamenatorze;
meore SemTxvevaSi
222
2ba
n+
= ; Sesabamisi anaTvali m2 egzamenatorze.
danayofis raodenoba egzamenatorze 12' mmm −= ;
danayofis raodenoba Tarazoze 21 nnn += ;
daxris kuTxe Tarazoze "n τ⋅ udris daxris kuTxes egzamenatorze
"' Эm τ , e.i. Эmn ττ '"=⋅ aqedan
'" "Эmn
τ τ= .
110022
nax. 105
α
Tu am moqmedebas gavimeorebT ramdenimejer, gamonaTvalTa saSualo
iqneba "τ -s zusti mniSvneloba. miRebul "τ gavyofT Suaze da SevitanT
zemoTaRniSnul formulebSi
2")("
2ττα baba
+=+
= ,
2")2("
22 ττα bakbak
−−=−−
= ,
amas ewodeba Tarazos RerZis daxris kuTxis gansazRvra naxevar
safasurebSi.
a) Tarazos mgrZnobiaroba
Tarazos mgrZnobiaroba aris mxilebis unarianoba, romelsac
gamoavlens Tarazo misi RerZis ama Tu im kuTxiT daxris SemTxvevaSi.
vTqvaT α kuTxiT daxril AB sibrtyeze devs ori Tarazo. erTi
gadaxrilia om manZi-
liT, meore ki '' mo -iT.
nulidan buStulis
centri daxrilia α
kuTxiT orive Tara-
zoze, magram rkali
erTi da igive kuTxi-
saTvis sxvadasxvaa.
meore Tarazos rkali ufro didia, e.i. is ufro mgrZnobiarea.
1Romρα
= ; 2'' Rmoρα
= ; ρα-kuTxuri radianuli zomaa.
Tarazos mgrZnobiaroba pirdapir proporciul damokidebulebaSia
radiusTan. Tarazos mgrZnobiaroba damokidebulia agreTve minis Siga
pireulis damuSavebaze, siTxiani buStulis sidideze da Tarazos
safasurze. Tarazos safasuri rac ufro mcirea, is miT ufro mgrZnobiarea,
e.i. Tarazos mgrZnobiaroba safasurTan ukuproporciul damokide-
bulebaSia. magaliTad, naxazze,
110033
nax. 106
31ατ = , xolo
42ατ = ;
mgrZnobiarobis mixedviT Tarazoebi iyofa 3 jgufad
1. zemgrZnobiare "10"2 −=τ -mde;
2. mgrZnobiare "30"10 −=τ ;
3. teqnikuri "120"30 −=τ .
b) xazebisa da sibrtyeebis horizontul mdgomareobaSi
moyvana da Tarazos Sesworeba
Tarazo wesrigSia maSin, rodesac Tarazos nuli aucileblad
Tarazos SuaSia, e.i. geometriuli RerZi AB paraleluria HH Tarazos
RerZis. aviRoT planSeti,
romelsac aqvs sami amwevi
xraxni A, B, C. davdoT
Tarazo ori amwevi AB
xraxnis gaswvriv. planSeti
rom Tarazul mdgomareobaSi
moviyvanoT, jer AB xazi
unda moviyvanoT Tarazul
mdgomareobaSi. AB-s gaswvriv dadebul Tarazoze buStulis SuaSi
mosayvanad A da B xraxnebs vabrunebT, sanam m wertili ar daemTxveva
nuls. Semdeg Tarazos SemovabrunebT ο180 da Tu Tarazo wesrigSia,
buStulis m centri daemTxveva O centrs da AB xazi iqneba Tarazulad.
xolo, roca Tarazo wesrigSi ar aris, buStulis Suaguli iqneba 'm
wertilSi. maSin 'mm -is (gadaxris) naxevars SevasworebT Tarazos
Semasworebeli C xraxniT, xolo meore naxevars gadavaadgilebT ukan amwevi
xraxnebiT AB da a. S.
cnobilia, rom sibrtyis mdebareobas sivrceSi gansazRvravs masze
mdebare ori xazi, amitom marto AB xazis moyvaniT Tarazul mdgomareobaSi
sibrtyis Tarazulobas ver mivaRwevT. amitom vdebT Tarazos AB xazis
110044
nax. 107
marTobulad C xraxnis mimarTulebiT da am xraxniT mogvyavs buStulas m
wertili nul wertilSi.
g) Tarazos RerZis Sesworebis geometriuli interpretacia
vTqvaT AB aris Tarazos
geometriuli RerZi, romelic
paraleluri unda iyos Tarazos
H1H2 RerZisa. davuSvaT, rom isini ar
arian paralelurebi da maT Soris
Seiqmna ε kuTxe, maSasadame,
rodesac Tarazos buStulas m
wertils nul wertilSi moviyvanT,
rac imas niSnavs, rom H1H2 RerZi
Tarazuli iqneba, maSin AB RerZi
daxrili iqneba ε kuTxiT. Tarazos
ο180 -iT Semobrunebisas buStula gadaixreba, radgan H1H2 RerZi miiRebs
'' 21 HH mdebareobas da gadakveTavs H1H2 RerZs β kuTxiT. naxazidan εβ 2= ;
gavyoT β< Suaze da gavataroT "" 21 HH RerZi, romelic
perpendikularulia AB -si; '' 21 HH RerZs, rom "" 21 HH -is mdgomareoba
mivaRebinoT, saWiroa igi Seswordes Tarazos Semasworebeli xraxniT 2βε =
kuTxiT. AB geometriul RerZs moviyvanT Tarzul mdgomareobaSi amwevi
xraxnebiT, ris Sedegadac AB paraleluria "2
"1 HH da orive RerZi
Tarazulia.
d) Tarazos RerZis daxris kuTxis gansazRvra
110055
nax. 108
H1
H '1
vTqvaT AB RerZs aqvs daxra α kuTxiT. Tu AB da H1H2 RerZebs Soris
ar aris ε kuTxe, maSin pirdapir miviRebT AB RerZis daxris kuTxes α -s,
magram rogori sizustis
Tarazoc ar unda iyos, ε -i
mainc iqneba (laparakia
uzustes Tarazoebze), amitom
εθα −= 1 . Semobrunebis Semdeg
(meore SemTxvevaSi)
εθα += 2
da maTi saSualo
221 θθ
α+
= sadac
( )2111τθ ba += ,
( )2222τθ ba += .
IX Tavi
samiznebeli xelsawyoebi
110066
nax. 109
ZiriTadi samiznebeli xelsawyo aris
adamianis Tvali. Cven gvainteresebs adamianis
Tvalis broli, sadac sxivebi gadaikveTebian
da badura, sadac gamosaxuleba miiReba.
davuSvaT, rom saWiroa AB sagnis
damzera, sadac:
α -saukeTeso xedvis kuTxea;
ω -saukeTeso xedvis manZili. igi sxvadasx-
vaa da tolia –
Cveulebrivi TvalisaTvis=25 sm.;
beci TvalisaTvis =10 sm;
yuSi TvalisaTvis =50 sm.
sagnis kargad garCevis mizniT usasrulod misi miaxloeba
SeuZlebelia. rogorc naxazidan Cans, Tvalis gugaSi (MM) miviRebT AB-s
Sebrunebul ba gamosaxulebas, romelsac vmzerT saukeTeso mxedvelobis ω
manZilze.
kuTxes, romelsac qmnis Tvalis optikur centrTan samzeri sagnis
kiduri wertilebidan wamosuli sxivebi, mxedvelobis kuTxe ewodeba (α ).
geodeziaSi Cven saqme gvaqvs Soreul sagnebTan. sagani rac ufro
Sorsaa, miT ufro mcire gveCveneba. saganTa garkveviT aRqma ki
aucilebelia. sagnebi, rom garkveviT aRviqvaT, saWiroa mxedvelobis kuTxis
gadideba e.i. Tvalis SeiaraReba. iaraRSi gamoiyeneba linzebi.
9.1. Wogri
Wogri Sedgeba obieqtivis da okularis muxlisagan.
110077
ganvixiloT, Cvens mier
Seswavlili, linzebis Tvise-
bebi rogor gamoiyeneba WogrSi,
romelic Soreul sagnebs
aaxloebs da adidebs. e.i. viRebT Sebrunebul warmosaxviT da gadidebul
gamosaxulebas.
Wogris obieqtivSi gamoiyeneba I SemTxveva,
okularSi ki II SemTxveva;
e.i. obieqtividan miRebul ba gamosaxulebas (Semcirebuls,
Sebrunebuls, namdvils, erTmag safokuso manZils gareT) davuxvedroT meore
linza, romelic mogvcems gamosaxulebas imave mxares-gadidebuls da
pirdapirs.
kepleris Wogri aaxloebs da adidebs, mikroskopi ki mxolod adidebs.
Wogri xasiaTdeba 3 TvisebiT:
1. gamadideblobiT,
2. mxedvelobis areTi,
3. naTelobiT.
a) Wogris gamadidebloba
Wogris gamadidebloba aris im kuTxis fardoba, romlis
farglebSic daimzireba sagani SeiaraRebuli TvaliT, kuTxesTan, romlis
farglebSic daimzireba sagani SeuiaraRebeli TvaliT, e.i. roca gvaqvs
Wogri da sagans vmzerT raRac β kuTxiT, xolo uWogroT α kuTxiT.
gamadidebloba αβ
=G , saerTod Wogris gamadidebloba rom gamoviyvanoT,
unda ganvsazRvroT obieqtivisa da okularis gamadidebloba cal-calke
da Semdeg maTi SeTavsebiT miviRoT, Wogris saerTo gamadidebloba.
110088
nax. 110
AB sagani obieqtiviT daimzireba α kuTxiT. miRebul ba
gamosaxulebas davuxvedroT Tvali saukeTeso xedvis ω manZilze, romelic
imave gamosaxulebas damzers 'α kuTxiT.
F – iyos mTavari safokuso manZili,
d – safokuso manZili linzidan gamosaxulebamde.
linzis mTavari fokusi aris is wertili, sadac miiReba
gamosaxuleba, rodesac sagani usasrulobaSia.
obieqtivis linzis gamadideblobis 1'g α
α⎛ ⎞=⎜ ⎟⎝ ⎠
formula gavamravloT
d-ze da ganvsazRvroT (d-F).
FDdFd ⋅=− ; aqedan
Dd unda SevcvaloT, amitom 1
'g αα
⎛ ⎞=⎜ ⎟⎝ ⎠
gavamravloT D-ze
FDF
Dd
−= , (9.1.1)
FDFF
DdFd
−=⋅=−
2
. (9.1.2)
F – aris mTavari safokuso manZili. igi SeiZleba iyos max 8-10sm. D
– aris manZili Wogridan sagnamde da rogorc zemoT avRniSneT 200≈ m. d –
manZili gamosaxulebidan linzamde da igi F≈ (radgan sagani Sorsaa).
axla ganvsazRvroT ab; naxazze abcΔ da 'abcΔ dan
2'2
22 αωα tgFtgab == . (9.1.3)
110099
nax. 111
arsebuli sidideebi mcirea da gamovsaxoT radianebSi.
'ωαα =F , aqedan (9.1.4)
gF==
ωαα '
. (9.1.5)
esaa obieqtivis linzis gamadidebloba da igi udris mTavari
safokuso manZilisa da saukeTeso xedvis manZilis fardobas.
b) ganvsazRvroT gamadidebloba okularisaTvis
vTqvaT, sagani aris fo-
kussa da linzas Soris. rom
ar gvqondes linza, sagans
davmzerdiT α kuTxiT, magram,
radgan linza gvaqvs, vmzerT
β kuTxiT. e.i. okularis
gamadidebloba αβ
=2g . unda
ganvsazRvroT 'D . (5)-dan
fDffd−
=−=−−'
2
ω aqedan
⎟⎠⎞
⎜⎝⎛ −=
ωffD 1' (9.1.6)
7max =f mm. 500=ω mm (yuSi TvalisaTvis) e.i. ωf SegviZlia
vugulvebelvyoT, e.i. fD ≈' . 'D - vayenebT safokuso manZilze (maSasadame
gamosaxuleba miiReba mTavar fokusSi). axla gavigoT ab. naxazidan
2'22 2
αωβ tgftgab == ; (9.1.7)
'ωαβ =f , 2'g
f==
ωαβ
, e.i. f
g ω=2 . (9.1.8)
(8) aris muSa formula okularis gamadideblobis gamosasaxavad.
axla ganvsazRvroT, rogor miiReba sxivebi WogrSi (obieqtivSi da
okularSi erTad).
111100
sagani AB gaivlis obieqtivis linzaSi. gamosaxuleba ba miiReba
Soreuli sagnis Sebrunebuli da Semcirebuli. axla davuxvedroT meore
(okularis) linza ise, rom ba iyos okularis linzisa da mis mTavar
safokuso manZils Soris. Wogris saerTo gamadidebloba
1 2F FG g g
f fω
ω= ⋅ = ⋅ = , (9.1.9)
e.i. gamadidebloba udris obieqtivis linzisa da okularis linzis
safokuso manZilebis fardobas.
nax. 112
2βaFtgab = , βα fF = , G
fF
==αβ
. (9.1.10)
gamadideblobas praqtikulad gansazRvraven sxvadasxva xerxiT.
ganvixiloT galileis meTodi:
Wogridan 20-25m-is manZilze ayeneben lartyas da mzeren
cali TvaliT WogrSi, meoreTi uWogrod. Tvalebi ise unda iyos
dayenebuli, rom orive suraTi erTmaneTs hfaravdes. daTvlian im
danayofebis ricxvs uWogrod, romelic Seesabameba WogrSi
lartyis 1 danayofs. igi Cvens SemTxvevaSi =12 e.i. 12=G , an
obieqtivis linzis Tavisufal ganxmulobas (xvrets) gazomaven
mm-Si, vTqvaT 30 mm. maSin Tvlian, rom gamadidebloba 30=G .
axla ganvsazRvroT rodisaa gamadidebloba maqsimaluri
da minimaluri.
111111
nax. 113
vTqvaT gvinda Wogrs hqondes maqsimaluri gamadidebloba,
usasrulobidan wamosuli sxivebi ikveTeba mTavar safokuso manZilze. igi
(F) obieqtivisaTvis mudmivi sididea.
okulari davuxvedroT f1, f2, f3 manZilebze. linzis zRvruli sidide
iqneba q (Tvalis gugis diametri), igi Tvalze didi ar SeiZleba iyos.
eqsperimentalurad dadgenilia, rom 7.5f : mm-ze naklebi ar
SeiZleba iyos, e.i. maqsimaluri gamadidebloba iqneba, rodesac f
minimaluria.
5.71max
FfFG == , (9.1.11)
FF
fFG
Qq =
⋅==
3min . (9.1.12)
Q – Tavisufali ganxmuleba obieqtivis linzisaTvis,
q – Tavisufali ganxmuleba TvalisaTvis.
9.2. mxedvelobis are
Wogris mxedvelobis are ewodeba sivrcis im nawils, romelsac
moicavs Tvali WogriT saganze mzeris dros mis uZrav mdgomareobaSi.
111122
nax. 114
mxedvelobis aris sazomi aris α kuTxe, romlis wvero obieqtivis
linzis optikur centrSia da gverdebi eyrdnoba ZafTa badis ganxmulobas.
mag. Tu mocemul sagans vmzerT obieqtivis linziT miviRebT
Sebrunebul, Semcirebul gamosaxulebas. mxedvelobis are
GGF
fFP '2000'2063'34386.0' ≈=⋅=⋅= ρα
P-s iReben 0.6-s f-isas (okularis linzis mTavari safokuso
manZilisa). praqtikulad mxedvelobis are ase ganisazRvreba
ρα ⋅=Dλ' . (9.2.1)
CD – mxedvelobis areSi ver moxvdeba.
9.3. gamadidebloba
miRebuli odenobebi saSualebas gvaZlevs gamoviTvaloT Wogris
gamadidebloba, mxedvelobis are da naTeloba, roca mas dayenebuli aqvs
hiugensis okulari, obieqtivSi araferi Secvlila, maSasadame miRebuli
formulebis gamoyenebiT gveqneba:
GFf
F
fGh
h
k
kep
k
k= = =
23 2
3 e.i. Semcirdeba. (9.3.1)
111133
mxedvelobis are h kh
k
2000© 2000 32 23
G Gα α= = = _ gadidda. (9.3.2)
fardobiTi naTeloba h Qq G
Q
q Ghh
h2
k2
k= = =085 085 49
94
2
2
2
2. . (9.3.3)
ganvixileT sxivebis svla hiugensis okularSi da miviReT
gadidebuli gamosaxuleba.
a) Wogris samzerad dayeneba
es procesi Sedgeba 2 moqmedebisagan:
1. Wogris Tvalze dayeneba;
2.Wogris saganze dayeneba.
1. saerTod mTavaria, rom yovelTvis naTlad vxedavdeT ZafTa badis
samanZilo Zafebs. igi moTavsebulia k1k2-anu hiugensis okularis minebis
SuaSi.
Zafebi rom naTlad davinaxoT, unda vamoZraoT okularis muxli, sanam
naTlad ar gamoCndeba Zafebi.
2-re moqmedebis dros xdeba Wogris saganze dayeneba. aq SeiZleba
Segvxvdes Semdegi SemTxvevebi:
I – normalur mdgomareobaSi
111144
nax. 115
ZafTa badis sibrtye emTxveva gamosaxulebis sibrtyes. e.i. saidanac ar
unda SevxedoT k-s igi yovelTvis dafaravs (2) SemTxvevas.
II-rodesac ZafTa badis sibrtye ar emTxveva gamosaxulebis
sibrtyes da adgili aqvs paralaqss (3 da 4).
nax. 116
aseT SemTxvevaSi t kramalieris saSualebiT unda vamoZraoT ise, rom
ZafTa badis sibrtye davamTxvioT gamosaxulebas, e.i. paralaqss ar hqondes
adgili.
b) Wogris optikuri Zala
TG
=ε"
(9.3.4)
ε -aris kritikuli kuTxe ≈ 60" . 60" -aris kuTxe, romliTac daimzireba
wertili (ω = 25sm) saukeTeso mxedvelobis manZilze.
111155
nax. 117
nax. 118
X Tavi
danayofebis amTvleli xelsawyoebi
10.1 anaTvlebis aRebis meTodi
geodeziuri samuSaoebis Sesrulebis dros saWiroa kuTxeebis
gazomva. kuTxeebi rom gavzomoT, unda vicodeT anaTvlis aReba. ganvixiloT
rogor xdeba anaTvlis aReba.
vTqvaT, gvaqvs rkali.
erTi danayofis Sesabamisi
centraluri kuTxe, avRniSnoT
λ-iT, maSin
1 – rkalisaTvis λ = 30'
2 – rkalisaTvis λ = 20'
Tu isari gaCerda
zustad danayofze, rogorc es
1-rkalzea naCvenebi, maSin
anaTvali β = 22 30ο ' , magram vTqvaT isarma gadainacvla kidev danayofis 0.7-
iT e.i. anaTvali iqneba
22 30 ' 21' 3' 22 51' 3'β = + ± = ±o o
aseve 2-re rkalisaTvis, vTqvaT isari gadascda danayofis 0.9-iT, maSin
anaTvali 22 40 ' 18' 2 ' 22 58' 2 'β = + ± = ±o o .
orive SemTxvevaSi anaTvlebi miaxloebiTia, e.i. anaTvlebs zustad ver
viRebT, amitom holandielma mecnierma
vernierma da portugalielma noniusma
gamoiyenes xelsawyo, romelsac ewodeba
vernieri.
adamianis Tvals
zedmiwevniT SeuZlia
gaarCios mijriT daye-
nebuli Skalebis dapi-
rispirebuli Strixebis
Tanxvedra. am Tanxved-
ras 10-jer ufro
111166
nax. 119
zustad arCevs, vidre grafikuli sizustea. vernierSi swored Tvalis es
Tvisebaa gamoyenebuli.
aviRos rkali AB da davyoT igi n nawilad, imave rkalis Sesabamisi
sigrZe wredze ki n-1 nawilad.
I-dan )1( −=⋅= nnAB λυ
Cveni mizania ganvsazRrvoT danayofebs Soris sxvaoba.
nλ
λ +=−= υτ (10.1.1)
II-re SemTxvevisaTvis, AB-
rkalis sigrZe verni-erze isev
n nawilad davyoT, xolo
Sesaba-misi rkalis sigrZe
wredze n+1 nawilad.
maSasadame II-dan
AB n n= ⋅ = +υ λ( )1 , aqedan
τ υ= − = −λλn
I-SemTxvevaSi gvaqvs dadebiTi vernieri,
II-SemTxvevaSi gvaqvs uaryofiTi vernieri.
radgan sizuste I – SemTxvevisaTvis dadebiTia, am verniers ewodeba
dadebiTi vernieri, agreTve uwodeben pirdapirs, imitom rom anaTvlebi,
rogorc vernierze, aseve limbze izrdeba erTi mimarTulebiT.
II-re naxazze, gvaqvs uaryofiTi vernieri, imitom, rom τ miviReT (_)
niSniT. Sebrunebuli imitom, rom warwerebi limbze da vernierze izrdeba
urTierT sawinaaRmdego mimarTulebiT.
saerTod instrumentebSi gvxvdeba I-li saxis vernieri. II-re saxis
vernierebi SeiZleba Segvxvdes saxazavebze.
111177
XI Tavi
gazomvebi da xazebis dasarva
11.1. pirdapiri gazomvebi
ricxvi aris materialuri samyaros raodenobrivi gamosaxuleba. igi
miiReba gazomvebis an gamoTvlebis Sedegad.
cnobilia, rom geodeziuri samuSaoebis Sesrulebis dros manZilebs
zomaven pirdapiri (meqanikuri) an arapirdapiri (analizuri da
instrumentuli) xerxiT.
praqtikis moTxovnis Sesabamisad, zemoxsenebuli gazomvebi sruldeba:
umartivesi instrumentebiTa da xerxiT; saSualo sizustis pirdapiri
xerxiT; maRali sizustis pirdapiri xerxiT; geometriuli da fizikuri
manZilmzomebis saSualebiT.
umartivesi xerxiT manZilebis gasazomad iyeneben: drois aRricxvas;
Tvalzomvas; adamianis bijs; sazom borbals; samiwaTmzomelo fargals da
sxva.
saSualo sizustis pirdapiri xerxiT manZilebis gasazomi iaraRebia
sazomi kverTxebi, sazomi bafTa da ruleti.
maRali sizustiT pirdapiri xerxiT manZilebs zomaven invaruli
(iderin giliomis) mavTulebiT da aRweven 61 10−⋅ da met sizustes. am
xerxiT manZilebis gazomvas sWirdeba didi dro da, agreTve, gauval
adgilebSi misi gamoyeneba SeuZlebelia.
geometriuli da fizikuri manZilmzomebiT manZilebis gazomvebi
sruldeba nebismieri saxis reliefis pirobebSi. geometriul manZilmzomebs
warmoadgens Zafebiani da optikuri manZilmzomebi (paralaqsuri
manZilmzomebiTurT), romelTa Teoria da praqtika dafuZnebulia
geometriuli optikis kanonebze. es meTodi meqanikuri xerxiT manZilebis
gazomvaze swrafia, mxolod gamoiyeneba SedarebiT mokle (erT
kilometramde) manZilebis gasazomad da agreTve xasiaTdeba mcire 42 10−⋅
sizustiT.
fizikur manZilsazomebSi iyeneben sinaTlis interferencias, bgeriTi
rxevebis gavrcelebis talRebs da eleqtromagnitur energias, e. w.
signalebis saxiT. bgeriT manZilsazoms mcire sizustis gamo geodeziaSi ar
iyeneben. rac Seexeba sinaTlis interferencias, anu sinaTlis
111188
ganaTebulobis gaZlierebis da Sesustebis movlenas, geodeziaSi
SezRudulad gamoiyeneba. aq sinaTlis miRebuli gamosxivebis talRebis
sigrZeebSi izomeba mcire zomis etaloni, romelsac e. w. optikuri
gadamravlebis gziT TanamimdevrobiT adareben grZel manZilebs. sinaTlis
interferenciis meTodiT geodeziaSi xdeba invaruli mavTulebis
komparireba da sakontrolo bazisebis gazomva. misi sizuste aRwevs 71 10−⋅ ,
magram am meTodiT jerjerobiT ver xerxdeba 900 metrze meti manZilebis
gazomva, agreTve gazomvaze ixarjeba didi dro; fizikur manZilsazomebSi,
iyeneben ra eleqtromagnitur energias, awarmoeben im drois gazomvas, rac
sWirdeba am energias (signals) gaiaros gaorkecebuli gasazomi manZili;
amave dros, Tu am energiis siCqare cnobili iqna, gasazomi manZilebi martivi
damokidebulebebiT isazRvreba.
fizikuri manZilsazomebi amJamad warmoidgineba: eleqtrooptikuri anu
sinaTlis manZilsazomebis, geodeziuri radiomanZilsazomebisa da
radiogeodeziuri sistemebis saxiT.
eleqtrooptikur, anu sinaTlis, manZilsazomebSi eleqtromagnitur
energias (signals) warmoadgens eleqtrooptikuri talRebis nakadi,
romelsac iReben moZravi eleqtrosadgurebisagan.
geodeziuri radiomanZilsazomebis eleqtromagnitur energias
(signals) warmoadgens radiotalRebi, romelTac iReben akumulatorebis
saSualebiT.
radiogeodeziuri sistemebi iqmneba zemoxsenebuli fizikuri
manZilsazomebis safuZvelze. amisaTvis ramdenime punqtze awyoben
signalebis gadamcem da mimReb danadgarebs, rac saSualebas iZleva
Tanadroulad ganisazRvros aRniSnul punqtebs Soris manZilebi; agreTve
igi gamoiyeneba aerofotoTviTmfrinavisa da dedamiwis xelovnuri
Tanamgzavris mdebareobis dasadgenad da sxva.
fizikuri manZilsazomebis Teoria da praqtika emyareba fizikuri
optikis, eleqtroteqnikis, eleqtronikisa da radioteqnikis ZiriTadi
cnebebis srulyofilad gamoyenebas, am SromaSi maT ar ganvixilavT.
aRniSnuli sakiTxebi avtoris mier ganxilulia calke
saxelmZRvaneloSi - „fizikuri manZilsazomebi“ – 1996 weli da damxmare
saxelmZRvaneloSi–„Tanamedrove geodeziuri xelsawyoebi“ – 2004 weli.
111199
Cven ZiriTadad aq ganvixilavT pirdapiri (meqanikuri) da arapirdapiri
(analizuri da instrumentaluri) xerxis gazomvebs.
sazomi aris is, riTac vzomavT. xazebis gasazomad sargebloben
foladis bafTiT da CxirebiT. Ffoladis bafTa aris 20-24-48 metriani.
Ggeodeziur samuSaoebSi yvelaze Sromatevadi da sapasuxismgebloa
xazebis gazomva.
nax. 120
Cxirebi unda iyos miduRebuli da ara -ase. vsargeblobT 20
metriani bafTiT, magram sinamdvileSi bafTa ar aris 20m. igi an metia an
naklebi. saWiroa misi namdvili sigrZis dadgena – anu komparireba.
Kkomparirebas awarmoeben: 1) stacionaluri, 2) savele komparatoriT.
komparatoris sigrZes iReben 21 an 22 metrs, Mmets vidre bafTaa. λ-
komparatoris sigrZe winaswar dadgenilia. (mas amowmeben normaluri
bafTiT)
a= λ + (mj-mc) [ ]20000 ( 6) ( 4) 20,010= + + − − = m,
gazomili bafTa gamovida 10mm-iT grZeli. vTqvaT, es gamocda
vawarmoveT t = 15ο –ze. Semdegi gazomvebis dros mxedvelobaSi unda miviRoT
miRebuli gansxvaveba da garkveuli temperaturuli koeficienti.
vTqvaT, aseTi bafTiT gavzomeT n – jer, e.i. gazomil xazs unda
davamatoT (n × 10)mm. xolo, roca moklea unda gamovakloT.
NN
nax. 121
ganvixiloT magaliTi: vTqvaT, gvinda gavzomoT xazi, gvyavs wina muSa,
ukana muSa da samuSaoTa mwarmoebeli. wina muSas aqvs 10 Cxiri, xolo
112200
ukanas 1 - Cxiri. wina muSas ukana muSa ayenebs TavisTavsa da saris
gaswvrivobaSi.
wina asobs, ukana agrovebs. ukanasknel Cxirs rom Caasobs wina muSa,
ukanas eqneba 10 Cxiri, maSin samuSaoTa mwarmoeblis kontrolis qveS ukana
muSa gadascems wina muSas 10 Cxirs, rac =200m.
vTqvaT, ukanasknel adgilze varT da aweria 3. aucileblad bafTa
unda gadavabrunoT da warwera 3 unda gavakontroloT. 3-is sapirispirod
warwera unda iyos 17; 6-14; 8-12 da a.S.
vTqvaT, gvaqvs gazomili:
bafTa 20 gadatana, 8-Cxiri, 6m. 4dm. 8sm.
400 + 160.00 + 6 + 0.40 + 0.08 = 566.48m
xazis bolos wina muSa miadebs 0-s da aTvlas awarmoebs
sawinaaRmdego mimarTulebiT, Tu xazis gagrZeleba ar SeiZleba.
Ggazomili xazis sigrZe L= 566.48 gamovida, magram es sigrZe swori araa,
radgan sinamdvileSi bafTa grZeli iyo e.i. namdvili sigrZe
0 120LL Lδ ε= ⋅ + + ; (11.1.1)
02048.566 LL =+⎟
⎠⎞
⎜⎝⎛ + δ ,
axla temperaturisaTvis α = −( )t t Lο ο
zogadad L L Lο = + ±Δ ε1 (11.1.2)
sadac ε α1 = −( )t t Lο (11.1.3)
tο - bafTis komparirebis dros temperatura
t - muSaobis dros temperatura
ε1 - Sesworeba temperaturisaTvis
L – xazis sigrZe
λ- bafTis sigrZe
kidev Seitaneba Sesworeba
adgilis daxrisaTvis
L Lο = ⋅cosν (11.1.4)
22 sin2
L L Lννε = − =o o
(11.1.5)
nax. 122
112211
esaa Sesworebis zusti formula, sadac mxedvelobaSi miRebulia
adgilis daxra, magram Tu ν < 2ο-ze, maSin daxra mxedvelobaSi ar miiReba.
11.2. xazebis dasarva
AB xazis nebismier or wertilze gatarebul vertikalur sibrtyes
AB xazis gaswvrivoba ewodeba (nax. 123). am xazis L0 orTogonaluri
proeqciis gazomvisas sazomi xelsawyo mudam mis gaswvriv unda iyos
daWimuli. am miznisaTvis aucilebelia paloebiT damagrebuli A da B
wertilebis maxloblobaSi dasmuli sarebi iyos maT gaswvriv. reliefis
Taviseburobisa an didi sigrZis gamo xSirad saWiro xdeba xazis
gaswvrivobis dadgena (dasarva) anu damatebiTi sarebis dasma AB xazis
gaswvrivobaSi. roca xazis sigrZe aRemateba 150 metrs, vake adgilebSi
sarebis dasmas awarmoeben yovel 60-90 wyvil nabijze, borcvnar adgilebSi –
yovel 15_30 wyvil nabijze da mTagorian adgilebSi ki ufro mokle
manZilebze. im SemTxvevaSi, rodesac xazis dasarvis TandaTanobiTi
miaxloebis xerxs viyenebT an, roca xazis A da B wertili kuTxis wveros
warmoadgens da adgili mniSvnelovnad uswormasworoa, maSin garda
gaswvrivobisa, aucilebelia am wertilebSi sarebi zustad Sveulad iqnes
dayenebuli. ganvixiloT orive SemTxveva.
B wertilSi dasmuli sari (nax. 124) α kuTxiT misi daxris gamo AB
xazis gaswvriv ar imyofeba. reliefis Taviseburebis gamo Tu dasarvisa da
xazis gazomvisas gamoviyenebT am saris bolo B1 wertils, xazis
gaswvrivobidan gadavixrebiT ε kuTxiT. gaswvrivobis am Secdomis odenoba
misi simciris gamo gamoiTvleba formuliT:
nax. 123
112222
ε ρ β ρ ρ α β' ' sin ' ' cos sin= ⋅ =
⋅⋅ ≈ ⋅
⋅bcAb
bBAb
hL0
, (11.2.1)
sadac ρ = 3438' aris radiani.
rogorc vxedavT, saris β = 0ο da β = 180ο daxris SemTxvevaSi
gaswvrivobis Secdomas adgili ar eqneba. maSasadame, xazebis dasarvisa da
gazomvis dros unda vecadoT, rom yvela sari AB xazis gaswvrivobaSi iyos
dayenebuli. vTqvaT, L0 344= m; h = 2 m; β = 90ο ; α = 87ο , maSin (1) formuliT AB
xazis mimarTulebis Secdoma ε ' '≈ 1 . im SemTxvevaSi, roca adgilze
daniSnuli wertili kuTxis wveros warmoadgens da masze dayenebuli saris
ZirSi damizneba ver xerxdeba, maSin aucilebelia sari dayenebuli iyos
gaswvrivobasTan erTad Sveulad. agreTve TandaTanobiTi miaxloebisaTvis
gamoyenebuli sarebi, sanam maT xazis gaswvrivobaSi davamagrebT, yovelTvis
Sveulad unda davayenoT, radganac isini (an maTgan) sxvadasxva
wertilebidan (wertilebi) daimzirebian. sarebis, rogorc xazis
gaswvrivobaSi, ise Sveulad dasayeneblad, gamoiyeneba Sveuli.
saWiroebisamebr AB xazis gaswvrivobaSi saris dasma xdeba AB
mimarTebiT SveuliT, xolo saris Sveulad dasayeneblad igi damatebiT
unda gaswordes AB mimarTulebis marTobulad SveuliT. tlanqad saris
Sveulad dayeneba SeiZleba Sveulis gareSe imave wertilebidan saris
Ziridan Tavisaken TvaliT gadaxris SefasebiT.
11.3. xazebis dasarvis SemTxvevebi
nax. 124
112233
praqtikaSi gvxvdeba dasarvis ori ZiriTadi SemTxveva: pirveli,
roca saWiroa dasmuli ori saris mixedviT xazis gagrZeleba, da meore, or
dasmul sars Soris xazis dasarva. teqstSi mocemuli wertilebi
aRniSnulia A da B asoebiT, xolo damatebiTi sarebi arabuli cifrebiT
da maTi dayenebis Tanamimdevroba – indeqsebiani imave cifrebiT.
das muli ori s aris m i x ed v iT x azis g agr Z eleb a
aq ganixileba rogorc vake, ise uswormasworo da dabrkolebis
mqone adgilebis dasarvebi.
a) vake adgilze dasmuli ori saris mixedviT
xazis gagrZeleba
am SemTxvevaSi dasarvas awarmoebs erTi kaci (damsarveli). igi
Sordeba B sars daaxloebiT 60_70 wyvili nabijiT (95_100 metri) (nax. 125a)
da Tavis gverdze gadaweviT gadaxedavs B sars ise, rom igi faravdes A sars
mTel mis simaRleze. amiT damsarveli adgens AB gaswvrivobas, romlis
Sesabamis sibrtyeSi igi adgilze ayenebs 1-l sars, rwmundeba imaSi, 1-li sari
faravs Tu ara A da B sars, agreTve igi dayenebulia Tu ara Sveulad
(nax.125b). Tu samive saris gverdebi moqceul iqna erT Sveul sibrtyeSi,
mizani miRweulia da Tu es piroba ar aRmoCnda Sesrulebuli, 1-li saris
mdebareoba unda Seswordes. imave wesiT unda iqnes dasmuli 2, 3, 4, . . .
sarebi. aseTi meTodiT xazis gagrZelebas ewodeba dasarva Tavisaken.
nax. 125
112244
cxadia, binoklis gamoyenebiT dasmuli sarebi ufro zustad iqneba
dayenebuli AB-s gaswvriv.
igive amocana amoixsneba ufro zustad, swrafad Teodolitis
gamoyenebiT, mxolod aq saWiro iqneba damxmare muSa. amisaTvis Teodolits
vcentravT B wertilSi (nax. 126), momwesobaSi mogvyavs da Wogrs vumiznebT A
sars ZirSi. wred-alidadas SemovabrunebT 180ο-iT an Wogrs gadavitanT
zenitze da damxmare muSas (TanaSemwes) vasobinebT 1, 2, 3, me-4 sarebs yovel
60_70 wyvil nabijze daSorebiT. gaswvrivobaSi sarebis dasmas vamowmebT
ZafTa badis Sveuli Zafis saSualebiT, risTvisac Wogrs vamoZravebT misi
brunvis RerZis mikrometruli xraxnis saSualebiT vertikalur sibrtyeSi
saris Ziridan bolomde. Tu saWiroa 1, 2, 3, me-4... sarebis nacvlad Stativebis
dadgma (magaliTad, bazisis gazomvis dros), maSin Wogrs vumiznebT
Stativebis Tavis centrebs. rogorc vxedavT, aqac dasarva warmoebs
Tavisaken.
b) xeobebis erT mxareze dasmuli ori saris mixedviT
xazis gagrZeleba mis meore mxareze
nax. 127
nax. 126
112255
am SemTxvevaSi samuSaos asrulebs erTi damsarveli, risTvisac igi
xan erT ferdobzea da xan meoreze. magaliTad (nax. 127), 1-l sars asobs AB
gaswvriv; me-2 sars 1-l da B saris gaswvriv; me-3 sars me-2 da 1-lis
gaswvriv; me-4 sars me-2 da 3-is gawvriv; me-5-s me-4 da 2-is gaswvriv da me-6
sars me-5 da 1-li saris gaswvriv. saerTod, amgvar SemTxvevaSi, sizustis
gadidebis mizniT yovelTvis sxivi unda gadiodes gamoyenebuli sarebidan
rac SeiZleba erTi saris Tavsa da meoris ZirSi. magaliTad, me-6 saris
dasmisTvis sxivi gadis me-5 saris Tavze da 1-li saris ZirSi.
cxadia, TeodolitiT ganxiladi amocana ufro advilad da zustad
amoixsneboda TanaSemwesTan erTad. amisaTvis sakmarisi iyo B wertilSi
Teodolitis dayeneba da cnobili wesiT Tavisaken sarebis dasma (3, 2, 4, 6, 5,
1).
g) xazis gagrZeleba mcire dabrkolebis gadalaxviT
ganxilad SemTxvevaSi, sanam SesaZlebloba aris, vawarmoebT
dasarvas Tavisken (nax. 128). vTqvaT, B da A gaswvriv dasmul iqna 1-li sari,
Semdeg ekeris an sxva saSualebiT avRmarTavT marTobebs B da 1-l
wertilebSi, gadavzomavT orive marTobze λ manZils, romelic scildeba
dabrkolebas da vsvamT adgilze B1 da 11 sars; Semdeg maTi saSualebiT
daismeba me-21 me-31 da me-41 sarebi. ukanaskneli ori saridan avmarTavT isev
nax. 128
112266
marTobebs da gadavzomavT isev λ sigrZeebs, cxadia, dasmuli me-3 da me-4
sarebi iqneba A, B da 1-li saris gaswvrivobaSi.
11.4. or dasmul sars Soris xazis dasarva
aqac ganixileba rogorc vake da uswormasworo, ise garkveuli
dabrkolebis mqone adgilebis dasarva.
a) or urTierT uxilav wertils Soris maRlobis dasarva
ganxiladi amocana sruldeba ori an erTi damxmaris gamoyenebiT.
pirvel SemTxvevaSi (nax. 129) damsarveli dgeba mwvervalze,
daaxloebiT AB gaswvrivobis axlos ise, rom igi xedavdes orive wertils
da asobs sars 11 wertilSi; gzavnis mwvervalis axlos damxmareebs 11B da
11A mimarTebiT da asmevinebs sarebs 21 da 31 wertilebSi ise, rom isini
erTmaneTs xedavdnen. damxmareebi gadaadgileben damsarvels sariT maT
gaswvrivobaSi da asmevineben sars 12 wertilSi. Semdeg damsarveli
gadaadgilebs TavianTi sarebiT orive damxmares 12B da 12A mimarTebiT,
nax. 129
112277
romlebsac ayenebinebs sarebs 22 da 32 wertilSi da ase Semdeg. damxmareebi
yovelTvis urTierTs unda xedavdnen, iseve, rogorc wina SemTxvevaSi.
pirveli moqmedebebi sruldeba swrafad da tlanqad, xolo AB xazisadmi
miaxloebisas sjobs, Tu SesaZlebelia damsarvelma saris nacvlad
gamoiyenos Teodoliti. aRwerili wesiT gagrZeldeba samuSao. dasarva
damTavrebulad iTvleba, roca 1-dan me-2 faravs B-s da me-3 ki A-s; agreTve,
2-dan 1-li faravs me-3, xolo 3-dan 1-li me-2-s. mwvervalze sami saris dasmisa
da Semowmebis Semdeg amoiReba miwidan aRniSnuli sarebi da 1-l wertilSi
TeodolitiT (an TvalzomiT) damsarveli damxmareebis gamoyenebiT
awarmoebs ferdobebis dasarvas Tavisaken rogorc A, ise B wertilidan.
meore SemTxvevas aqvs adgili, roca erTi damxmare gvyavs da
Teodoliti ara gvaqvs, amave dros gansakuTrebuli pirobebis gamo saWiroa
erTbaSad AB-s gaswvriv oTxi, xuTi, eqvsi da meti saris dasma. vTqvaT, AB-s
gaswvriv saWiroa erTbaSad eqvsi saris dasma (nax. 130).
eqvsive sari unda iyos dasmuli daaxloebiT AB-s gaswvriv ise, rom,
garda pirvelisa da ukanaskneli (meeqvse) sarisa, danarCeni yoveli saris
dgomis wertilidan moCandes orive mxareze or-ori, xolo pirvelive da
ukanaskneli (meeqvse) saridan aucilebelia erT mxareze moCandes uaxloesi
mocemuli da meore mxareze dasmuli ori sari. ase, magaliTad, 11 wertilSi
dasmuli pirveli saridan: erT mxareze unda Candes A sari da meore mxareze
21 da me-31 sari , me-21 wertilSi dasmuli me-2 saridan erT mxareze 11 da A,
xolo meoreze 31 da me-4 sari da ase Semdeg 6 saramde., romlidanac Cans:
erT mxareze B, meoreze 51 da me-41 sarebi.
nax. 130
112288
dasarvis zogadi wesi mdgomareobs yoveli Sualedi saris
gadaadgilebaSi misi mosazRvre sarebis gaswvriv: ase, magaliTad, damsarveli
dgas ra me-21 wertilSi dasmul sarTan, 11 wertilSi dasmul sars
gadaataninebs damxmares 21A gaswvriv 12 wertilSi; mere TviT gadadis me-31
wertilSi da damxmares me-21 wertilidan gadaataninebs sars 31 12 gaswvriv
22 wertilSi; analogiurad gadaitaneba sari me-31 wertilidan 41 22-is
gaswvrivobis 32 wertilSi; me-41 wertilidan 51 32 gaswvrivobis 42 wertilSi;
me-51 wertilidan 61 42 gaswvrivobis 52 wertilSi, me-61 wertilidan B52
gaswvrivobis 62 wertilSi, riTac adgilze iqmneba Tokis A122232425262B
mravalkuTxedi. Tu davukvirdebiT, vnaxavT damsarveli yovelTvis dgeba
Sualedi gaswvrivobis sawyis wertilSi. saWiroa aRniSnuli Tokis
mravalgverdi gadavaqcioT wrfed. amisaTvis damsarveli gadadis me-62
wertilSi da damxmares gadaataninebs me-52 wertilidan sars 62 42
gaswvrivobis me-53wertilSi; analogiurad xdeba sarebis gadaadgileba: me-42
wertilidan 53 32 gaswvrivobis me-43wertilSi; me-32 wertilidan 43 22
gaswvrivobis 33 wertilSi; me-22 wertilidan 32 12 gaswvrivobis me-23
wertilSi. Semdeg 12 wertilidan 23A gaswvrivobis 13 wertilSi; me-23
wertilidan 33 13 gaswvrivobis me-24 wertilSi; me-3 wertilidan 43 24
gaswvrivobis me-34 wertilSi. analogiuri gziT daismeba sarebi me-44, 54, me-63,
. . . ,wertilebSi. rogorc vxedavT, Tokis mravalgverdi TandaTan
uaxlovdeba AB xazs da yvela sari iqneba AB-s gaswvriv, Tu nebismieri
Sualedi saridan mzeriT mis gverdze dasmuli sari dafaravs am
ukanasknelis Semdgom sars.
sizustis gazrdisa da muSaobis nayofierad Sesrulebis mizniT
saWiroa gamoviyenoT sarebis rac SeiZleba mcire raodenoba.
112299
b) xeobis orive mxareze dasmul sarebs
Soris xazis dasarva
am SemTxvevaSi saWiroa erTi damxmare. damsarvels uxdeba xeobis
orive mxares TanamimdevrobiT dgoma. igi A wertilSi dgomiT damxmares
daasobinebs 1-l da me-2 sarebs, xolo B wertilSi dgomiT ki me-3 da me-4
sarebs (nax. 131). cxadia, amgvari an analogiuri amocanebis amoxsnisaTvis
saukeTeso saSualebaa Teodolitis gamoyeneba. saerTod saWiroa
damatebiTi wertilis ise SerCeva, rom am wertilze gamavali sxivi gadiodes
gamoyenebuli sarebidan erTis ZirSi da meores boloSi. Tu reliefi
SedarebiT mSvidia, dasarva sruldeba Tavisaken (nax. 132).
nax. 132
nax. 131
113300
XII Tavi
zogierTi cnobebi ganazomTa
Secdomebis Teoriidan
cnobilma mecnierma, akademikosma a. aleqsandrovma SesaniSnavad
daaxasiaTa gazomvebi da maTi Secdomebis maTematikuri damuSavebis
problema: „Tanamedrove mecnierebaSi yoveli axali mimarTuleba ufro
xSirad warmoiSoba gazomvebis organizaciis axal meTodTan (tipTan)
kavSirSi: an ufro maRali mgrZnobiarobiT an sxvadasxva movlenebze
erTdrouli dakvirvebiT, Tanxvdenilni an acdenilni droSi, an erTi da
imave movlenaze dakvirvebiT sivrcis sxvadasxva wertilidan.
axali aRmoCenebis umravlesoba dakavSirebulia swored imasTan, rom
axleburad iqna Camoyalibebuli dakvirvebis meTodika, aseve axleburadaa
organizebuli TviT gazomvebis sistemebi, maTi Sedegebis damuSaveba...“
sazomi teqnikis sxva saxeebTan SedarebiT eleqtruli gazomvebi,
rogorc wesi, gamoirCeva didi sizustiT, simartiviTa da saimedoobiT. amis
gamo amJamad eleqtrosazomi xelsawyoebis meqanizmebi ixmareba agreTve
bevri araeleqtruli sididis gasazomad da sul ufro da ufro metad
gamoiyeneba sxvadasxva sawarmoo procesis kontrolisa da
avtomatizaciisaTvis.
unda ganvasxvavoT cnebebi „xelsawyos cdomileba*“ da „gazomvis
cdomileba“. es aixsneba imiT, rom gazomvis Sedegis sizusteze moqmedebs
ara marto sazomi xelsawyoebis cdomilebebi, aramed gazomvis procesSi
damatebiTi mowyobilobebisa da aparatebis gamoyenebiT gamowveuli
cdomilebebic. am mowyobilobebs miekuTvneba xelsawyos gazomvis zRvris
gazrdis saSualebebi – Suntebi da damatebiTi winaRobebi, agreTve sazomi
tranformatorebi. Tu calkeuli cdomilebebi erTi da imave niSnisaa,
gazomvis sizuste Semcirdeba. xolo, Tu cdomilebebis niSani sxvadasxvaa,
isini erTgvarad daakompensireben erTmaneTs.
* xSirad „cdomilebis“ nacvlad xmaroben „Secdomas“, rasac arsebiTad erTnairi
Sinaarsi aqvs.
113311
12.1 gazomvebis saxeebi
obieqtis saxeobisa da xasiaTis mixedviT geodezia – markSeideriaSi,
mimarTaven xazovan da kuTxur, agreTve haeris temperaturis da wnevis
gazomvebs.
gazomvebs yofen:
1. imis mixedviT Tu rogor aris miRebuli gasazomi obieqtis sidide –
pirdapiri (uSualo) da arapirdapiri (arauSualo, meSveobiT);
2. obieqtis gazomvaTa raodenobis mixedviT – aucilebeli da Warbi;
3. Tvisebebis mixedviT – erTgvarovani da araerTgvarovani;
4. pirobebis mudmivobis mixedviT, romelzec damokidebulia ganazomTa
sizuste – tolzusti da aratolzusti;
5. obieqtis elementTa ganazomebis urTierTdamokidebulebis mixedviT
– damoukidebeli da damokidebuli, pirobiTi da upirobo.
12.2 gazomvebis Secdomebi da Sesworebani
gazomvebis dros mxedvelobaSi unda gvqondes ori garemoeba;
1) sazomi xelsawyos gazomvis procesSi ucvleloba. imisaTvis, rom
sazomis mciredi cvalebadoba mxedvelobaSi iqnes miRebuli, saWiro xdeba
gazomvis dawyebamde da mis Semdeg Semowmdes sazomis sigrZe, agreTve
gazomvis procesSi garemos cvalebadoba aRiricxos (temperatura da sxv.)
da ganazomSi Setanil iqnes saTanado Sesworeba.
2) gasazomi obieqtis odenoba ar unda icvlebodes gazomvis
procesSi. bunebaSi absoluturad ucvleli obieqtebi ar arsebobs, magram
SeiZleba moiZebnos drois iseTi Sualedi, roca gasazomi sidide gazomvis
sizustis farglebSi miviCnioT ucvlelad. umaRlesi sizustiT
gansazRvrul raime sidides, romlis mniSvneloba drois garkveul manZilze
ucvleli rCeba, ewodeba namdvili, anu WeSmariti mniSvneloba gasazomi
sididisa. magaliTad, invaris mavTuliT gansazRvruli xazis sigrZe
WeSmaritia drois garkveul periodSi, imave xazis foladis bafTiT
gazomvis SedegTan SedarebiT.
113322
Tu raime sididis WeSmarit mniSvnelobas aRvniSnavT X-iT da imave
sididis gazomvis Sedegad miRebul mniSvnelobas il -iT, xolo WeSmarit
Secdomas δi-iT, maSin WeSmariti Secdoma ganisazRvreba gamosaxulebiT
i i Xδ = −l (12.2.1)
da WeSmariti Sesworebis odenoba, romelsac iε -iT aRvniSnavT, gantolebiT
i iXε = − l . ( 1, 2, 3...)i =l (12.2.2)
Secdomis odenobis dasadgenad g azom v iT m iRe bul ode nob a s
v a klebT i m a v e s ididis i m m n iS v n eloba s , rom elic Teoriuli
g a moTvliT a n uzuste s i g azom v i s m eTode b iT g v a q v s
m iR e buli , magaliTad, Tu SvidkuTxa mravalkuTxedis kuTxeTa jami
gazomvis Sedegad Seadgens 900 2 'o -s. Secdoma anu Seukvreloba iqneba 2 '+ ,
xolo Sesworeba – 2 ' . bafTis sigrZe miRebuli gvaqvs 20 metris tolad,
sinamdvileSi SemowmebiT miviReT, rom igi 20,008 metris sigrZisaa. am
SemTxvevaSi bafTis nominaluri sigrZis Secdoma iqneba +8 mm. xazis
gazomvisas aseTi bafTiT yovel gazomvaSi Secdoma – 8 milimetria, xolo
Sesworeba iqneba + 8 milimetri. Secdomisa da Sesworebis amgvarad
gansazRvra logikurad savsebiT gamarTlebulia cxovrebaSi miRebuli
wesis mxrivac. magaliTad, amboben – Secdoma moviciloT, ukuvagdoT da
Sesworeba miviRoT. maSasadame, SecdomiT miRebuli Sedegis Sesasworeblad,
saWiroa ganazoms Secdomis sidide algebrulad gamovakloT an Sesworeba
davumatoT. marTlac, (1) formulidan miviRebT,
i iX δ= −l , (12.2.3)
xolo (2) formulidan gveqneba
i iX ε= +l . (12.2.4)
rogorc vxedavT, sididis WeSmariti odenobis maTematikurad miReba,
ganazomisagan Secdomis gamoklebiT an Sesworebis mimatebiT xdeba.
Secdomis an Sesworebis niSnis arasworad gansazRvra gamoiwvevs
SedegSi Secdomis gaorkecebas, amitom zemoT miRebuli wesi mudam unda
gvaxsovdes.
Secdomasa da Sesworebas Soris gansxvaveba mxolod niSnebis
sxvadasxvaobaSi ar mdgomareobs, maT Soris gansxvaveba unda gvqondes
warmodgenili sxvagvaradac, magaliTad,
113333
1. bafTis damzadebis Secdomasa da Sesworebas Soris niSnebSia
gansxvaveba da maTi absoluturi sidideebi tolia. e.i.
δ ε= .
2. geodeziaSi Secdomis odenobas adgenen (1) formuliT. magaliTad,
pirobiT ganazomebSi Secdoma tolia uSualod Sesrulebul ganazomTa da
Teoriul gamonaTvalTa Soris sxvaobisa, Sesworebebi ki gawonasworebis
Sedegad miRebuli odenobebia: mxolod obieqtis elementTa Sesworebebis
absolutur odenobaTa jami tolia obieqtis pirobiTi ganazomis Secdomis
absoluturi odenobisa.
magaliTad, brtyeli samkuTxedis kuTxeTa jamis Secdoma (W)
gamoiTvleba gamosaxulebiT
( ) 180W α β γ= + + − o , (12.2.5)
kuTxeebis SesworebaTa jami ki formuliT
( )1 2 3 180ε ε ε α β γ+ + = − + +o , (12.2.6)
e.i.
1 2 3W ε ε ε= + + . (12.2.7)
amrigad, Secdomis absoluturi mniSvneloba tolia SesworebaTa
absoluturi mniSvnelobebis jamisa, anu SesworebaTa jami udris Secdomas
Sebrunebuli niSniT.
3. Sesworeba araa Sedegi Secdomis. magaliTad,
bafTis sigrZis Sesworeba temperaturis cvalebadobis gamo
( )0t t tε α ι= − , (12.2.8)
sadac α aris temperaturuli gafarToebis koeficienti;
t - temperatura xazis gazomvis dros;
0t - temperatura bafTis komparirebis dros ;
ι - komparirebiT gansazRvruli bafTis sigrZe.
gazomili xazis Tarazulobaze dayvana, anu Sesworeba xazis daxris
gamo ganisazRvreba formuliT
22 sin2
Lννε = , (12.2.9)
sadac L aris daxrili xazis sigrZe;
ν - xazis daxris kuTxe.
113344
am magaliTebidan Cans, rom Sesworebasa da Secdomas Soris
gansxvaveba arsebiTia.
12.3 ganazomTa rigebis saxeebi
ganazomTa rigebSi gansxvavebuli cifrebi ori saxisaa da praqtikaSi
Sesabamisad ganazomTa rigebic ornairi saxis gvxvdeba.
ganazomTa pirveli rigi iqneba im SemTxvevaSi, roca ganazomi
obieqtis mravaljer gazomvis Sedegebi icvleba raime garkveuli wesis
mixedviT.
meore saxis rigi ar xasiaTdeba wevrTa raodenobis kanonzomieri
cvalebadobiT.
12.4 ganazomTa Secdomebis saxeebi
ganazomTa romelime rigis miReba damokidebulia im komponentebze,
romlebic monawileobs gazomvebSi (gasazomi obieqti, xelsawyo, mzomi an
gamomTvleli da garemo pirobebi).
ganazomTa Secdomebs yofen:
warmomSobi wyaroebis mixedviT – xelsawyos, piradi, garemo pirobebiT
da obieqtiT gamowveuli; xasiaTis, anu TvisebaTa mixedviT – tlanqi,
sistematuri, mudmivi da SemTxveviTi.
ama Tu im bunebis, anu Tvisebis Secdomis warmomSobi mizezi SeiZleba
iyos gazomvis calkeuli faqtori an yvela faqtori erTad. amitomac
sainJinro – geodeziuri da markSeideruli gazomvebis Tanmxlebi
ucilobeli Secdomebis analizi da maTi saboloo Sedegebze Semoqmedeba
Seiswavleba specialur disciplinaSi, romelsac - geodeziur -
markSeideruli ganazomebis maTematikuri damuSavebis Teoria“ ewodeba
(n. TevzaZe).
cnobilia, rom yoveli ganazomis tlanqi, sistematuri da mudmivi
Secdomebisagan ganTavisufleba SeiZleba. rac Seexeba, SemTxveviT
Secdomebs, maTi mospoba Cven ar SegviZlia, am Secdomebs emateba narCeni
sistematuri da mudmivi Secdomebi. maTi erToblioba, romelsac
„ucilobeli“ Secdoma ewodeba, gvaZlevs meore saxis rigs.
113355
12.5 SemTxveviT SecdomaTa Tvisebebi
manZilis mravaljer gazomvis analizis safuZvelze dadgenilia, rom
SemTxveviTi Secdomebi garkveul pirobebSi tolzusti gazomvebis rigebSi
warmoiSoba normaluri statistikuri ganawilebis kanoniT. es
kanonzomiereba erTnairia nebismieri saxis gazomvisaTvis. aseT SecdomaTa
kanonzomiereba miT ufro mkveTria, rac ufro metia rigis wevrTa ricxvi.
es aris statistikuri kanonzomierebis zogadi Tviseba da mas did ricxvTa
kanons uwodeben.
manZilis mravaljer tolzusti gazomvis Sedegad miRebuli ganazomTa
meore saxis rigis safuZvelze dgeba SecdomaTa rigi, romelic iqneba meore
saxis da mas ewodeba SemTxveviTi xasiaTis ucilebel SecdomaTa rigi. igi
xasiaTdeba Semdegi TvisebebiT:
1. Secdomebis absoluturi odenoba ar aRemateba garkveul zRvars;
2. rac ufro didia Secdomebis absoluturi mniSvneloba, miT ufro
iSviaTad gvxvdeba rigSi;
3. absoluturad toli dadebiTi da uaryofiTi Secdomebi
daaxloebiT erTgvari raodenobiT gvevlineba rigSi;
4. SecdomaTa saSualo ariTmetikuli mcire sasruli odenobisaa da
gazomvaTa ricxvis usasrulo gadidebis SemTxvevaSi misi zRvari
nulis tolia.
12.6. gazomvis erTeulebi
gazomvis dros gansasazRvrav sidides adareben zomis erTeuls.
adre erTeulis SerCevis sakiTxi wydeboda nebismierad. pirveli erTeulebi
ukavSirdeboda adamianis sxeulis zomebs: futi (terfis sigrZe), saJeni
(manZili gaSlili xelebis boloebs Soris), diumi (cera TiTis sigane) da
a.S. mravali am erTeulTagani dRemde SenarCunebulia, rac qmnis aSkara
uxerxulobas sigrZeebis saerTaSoriso savaWro urTierTobebSi,
samecniero-kvleviT da saproeqto samuSaoebSi. amitom mecnierebma
gadawyvites daedginaT yvela qveyanaSi gamosayenebeli zogadi erTeulebi.
113366
radgan praqtikaSi saqme gvaqvs gasazomi sididis sxvadasxva
mniSvnelobasTan, amitom mizanSewonilia gvqondes Sesabamisad sxvadasxva
zomis erTeulebi, magram daculi unda iqnes piroba, rom erTi erTeulidan
meoreze gadasvla xvdebodes SeZlebisdagvarad swrafad. frangi
mecnierebisagan (borda, kondorse, laplasi, monJi) Semdgarma komisiam,
romelic 1790 wels Seiqmna, nacionaluri krebis dadgenilebis safuZvelze,
sigrZis erTeulad miiRo dedamiwis meoTxedis erTi meaTmilionedi nawili
da 1791 wels safrangeTSi am erTeulad SemoRebuli iqna metri. metris
prototips warmoadgenda platinis da iridiumis Senadnobisagan
specialurad damzadebuli kverTxi (saxazavi). es Senadnobi imitom iqna
arCeuli, rom mas aqvs siTburi gafarToebis metad mcire koeficienti da
koroziis mimarT mdgradia. bevrma qveyanam gadaiRo am kverTxidan asli
sakuTari zomiTi SedarebisaTvis.
me-19 saukunis meore naxevridan metruli sistema farTod gavrcelda
da dakanonda TiTqmis yvela qveyanaSi. zomaTa metruli, anu rogorc mas
xSirad uwodeben, aTobiTi sistemis gamorCeul Tvisebas warmoadgens is, rom
masSi erTi da imave sididis sxvadasxva erTeuli erTmaneTTan
dakavSirebulia aTis mTeli dadebiTi an uaryofiTi xarisxebiT.
Semdgom astronomiul-geodeziur gazomvaTa sizustis gazrdis
Sedegad aRmoCnda, rom sigrZis arCeul erTeulsa da misTvis damzadebul
prototips Soris arsebobs sakmao gansxvaveba, magram gadawyda, rom
daefiqsirebinaT es prototipi, rogorc sigrZis erTeulis ZiriTadi
etaloni, radganac ar iyo garantia imisa, rom axali dazusteba ar
gamoiwvevda mis xelaxal Secvlas. prototipis mixedviT metris
gansazRvrasTan dakavSirebiT daikarga metruli sistemis erT-erTi
upiratesoba – misi daculoba da zusti aRwarmoebis SesaZlebloba.
SemdgomSi SesaZlebeli gaxda metris sigrZis dakavSireba gansazRvruli
speqtruli xazis talRis sigrZesTan. am xazis saxiT miRebul iqna
kriptonis narinjisferi xazi. metris Tanamedrove gansazRvra Semotanilia
1960 wels, romlis Tanaxmad metri Seicavs vakuumSi kriptonis speqtruli
xazis talRis sigrZis 1650763,73-s, masobrivi ricxviT 86
8636
⎛ ⎞⎜ ⎟⎝ ⎠
kg. aTobiT
sistemaSi arsebobs sigrZis Semdegi erTeulebi: kilometri (1km)=103 m; metri
113377
(1m)=10dm=102 sm; decimetri (1dm)=10sm; santimetri (1sm)=10mm; milimetri
(1mm)=103mkm; mikrometri (1mkm)=103nm; nanometri (1nm)= 10Ao ; angstremi (1Ao )=10-10m.
erTeulTa saerTaSoriso sistemis (sast 9867-61) SemoRebamde, romelic
aRiniSneboda abreviaturiT CИ , mikrometri iwodeboda mikronad da
aRiniSneboda mk. zogjer, sxva erTeulTa rusuli aRniSvnebis dros,
mikroni aRiniSneboda berZnuli asoTi μ (miu), romelic Sedis
saerTaSoriso aRniSvnaTa erTobliobaSi, miuxedavad imisa, rom
saxelwodeba „mikroni“ da „mk“ gauqmebulia, igi arc Tu iSviaTad gvxvdeba
literaturaSi. nanometri adre iwodeboda milimikronad da aRiniSneboda
mmk.
farTobis erTeulad miiReba im kvadratis farTobi, romlis gverdi
sigrZis erTeulis tolia: 1km2=106m2; 1m2=104sm2; 1dm2=10-2m2=100sm2; 1sm2=100mm2;
1mm2=10-6m2. miwis zomis erTeulad miRebulia heqtari: 1ha=10-2km2=10ar=104m2.
moculobis erTeulad miiReba kubis moculoba im wiboTi, romelic
sigrZis erTeulis tolia: 1m3=103dm3=106 sm3, 1dm3=103sm3, 1sm3=103mm3.
kuTxur zomad gamoiyeneba gradusebi, minutebi da sekundebi. kuTxuri
gradusi aris wrewiris centraluri kuTxe, romlis rkali Seadgens erT
rkalur graduss, anu wrewiris 1/360 nawils. gradusi – 1 60 '=o ; minuti
1' 60"= . marT kuTxes zogjer yofen 100 nawilad. marTi kuTxis meased
nawils uwodeben gons (adre grads); goni – 1 100=d s ; meaTedi minuti –
1 100=s ss ; meaTedi sekundi – 1 10=ss -4 goni. dasaxelebul kuTxur zomebs
Soris arsebobs Semdegi Tanafardoba:
1 1,111...=o d ; 1 0,9= od ;
1' 1,851...= s ; 1 0,54 '=s ;
1" 3,08641075= ss ; 1 0,324"=ss .
erTi kuTxuri zomis meoreSi gadayvanis gasaadvileblad adgenen
cxrils.
rkalur zomaSi wrewiris centraluri kuTxe ganisazRvreba rogorc
rkalis sigrZis fardoba radiusTan – r
ϕ =l, sadac l rkalis sigrZea, xolo
r - radiusi.
113388
rkaluri zomis erTeulis mniSvnelobas Seesabameba ρ kuTxe,
romlisTvisac rkalis sigrZe radiusis tolia. rkaluri zomis am erTeuls
uwodeben radians. ρ kuTxe SeiZleba mocemul iqnas or kuTxur zomaSi:
360 / 2 57 ,3ρ π= ≈o o o ; /200 63,6πρ = =d d d ; ' 3438'ρ = ; 6366ρ =s s ; " 206265ρ = ;
636620ρ =ss ss .
CИ _ ZiriTad erTeulebs miekuTvneba agreTve droisa da masis
erTeulebi, drois erTeulad miRebulia wami (wm). drois erTeulTa
sistemaSi Sedis: wuTi – 1wT=60wm; saaTi – 1sT=60wT=3600wm.
masis erTeulad miRebulia kilogrami (kg) – es aris platinisa da
iridiumis Senadnobisagan damzadebuli etalonis masa. kilogramis
saerTaSoriso etaloni inaxeba q. sevrSi (safrangeTi). masis etalonis
mixedviT damzadebulia aslebi da gadacemulia sxvadasxva qveyanaSi.
aTobiT sistemaSi: tona – 1t=103kg; grami – 1g=10-3kg; miligrami – 1mg=10-3g.
litri (l), romelic xSirad iwodeba „tevadobis erTeulad“, adre
ganisazRvreboda rogorc moculoba, romelsac ikavebs 1kg wyali 4 Co -is
dros. es moculoba Seadgens 1, 000028 dm3. 1964 wels litri gatolebul
iqna erT kubur decimetrTan: 1l=1dm3. miwis masis moculobis
gansazRvrisas iyeneben ZiriTad erTeuls 1m3-s.
zemoT CamoTvlil erTeulTa safuZvelze Seqmnilia Zalisa da
rxevaTa sixSiris warmoebuli erTeulebi. Zalis erTeulad miRebulia
niutoni. erTi niutoni (1n) aris Zala, romelic 1kg masis mqone sxeuls
aniWebs 1m/wm aCqarebas Zalis moqmedebis mimarTulebiT. aTobiT sistemaSi
arsebobs kiloniutoni – 1kn=103n. adre Zalis erTeulad zogjer iyenebdnen
kilograms (1kg); 1kg Zala q. sevris ganedze daaxloebiT Seesabameba 9,81n.
f rxevebis sixSiris erTeulad miiReba herci (1hc) – es aris sixSire,
romlis drosac wamSi sruldeba perioduli sistemis erTi cikli. aTobiT
sistemaSi: kiloherci – 1khc=103hc; megaherci – 1mghc=106hc=103khc.
113399
nax. 133
XIII Tavi
kuTxzomiTi agegmva
vTqvaT, mocemulia sivrceSi A, B da C wertilebi da saWiroa β -
kuTxis orTogonaluri proeqciis gazomva.
arsebuli SemTxvevisaTvis igulisxmeba mcire sivrce da ufleba gvaqvs
davagegmiloT donebriv zedapirze. miviRebT A0, B0, C0 orTogonalur
proeqciebs. miRebuli β0 kuTxe aris orwaxnaga kuTxis proeqcia, romelic
izomeba Sesabamisi xazovani kuTxiT. AB da AC xazebis orTogonaluri
proeqciebi iqneba A0B0 da A0C0.
Tu BAC kuTxes davagegmilebT Tarazul sibrtyeze, miviRebT Tarazul
kuTxes, xolo Tu davagegmilebT vertikalur sibrtyeze miviRebT daxris
114400
kuTxes, im SemTxvevaSi, rodesac sivrceSi erT-erTi gverdi Tarazuladaa,
vertikaluri kuTxis kerZo SemTxvevaa daxris kuTxe.
A0A gavagrZeloT, davayenoT Sveuli, Tarazuli wrediT, romelic
dayofilia gradusebad. B da C wertilSi davayenoT sarebi da davmziroT
AB da AC mimarTulebebs Soris kuTxe.
β β β0 = −C B , (13.1)
marjvena mimarTulebis anaTvals akldeba marcxena mimarTulebis
anaTvali. vTqvaT, gvinda gavigoT ara β0 , aramed β0 ' (ix. nax. 208). Tu βB
mcirea βC -ze, maSin vamatebT 360ο da gveqneba
β β β0 360' ( )= + −B Cο . (13.2)
kuTxes vzomavT Semdegi wesiT: saxiT vdgebiT kuTxisaken, kuTxis
wveroze vayenebT kuTxmzom iaraRs, romlis limbi Tarazuladaa da viRebT
anaTvals orive mimarTulebaze, marjvenas vaklebT marcxenas, Tu marjvena
naklebia marcxenaze vamatebT 360ο -s.
daxris kuTxes vzomavT analogiurad, amisaTvis saWiroa gvqondes
Sveuli wredi da Tarazuli wredi. Tu A wertilidan gadavzomavT AB-s
proeqciis paralelurs, miviRebT daxris kuTxes ν -s, B wertilisaTvis
aRmavlobiT, radgan B – A-Tan SedarebiT maRlaa, xolo C wertilisaTvis
_ ν' -daRmavlobiT. kuTxeebs vzomavT kuTxmzomi iaraRiT – TeodolitiT.
Teodoliti idgmeba samfexze, (Stativze) da magrdeba xerxemala
xraxniT. mas aqvs 3 Sto, romelSic gayrilia amwevi an damyenebeli xraxnebi.
saerTod instruments aqvs 5 saxis xraxni:
1. xerxemala xraxni, romliTac magrdeba Teodoliti samfexze;
2. amwevi anu damyenebeli (3) xraxnebi;
3. damtkecni anu dammagrebeli (limbze, wred-alidadaze);
4. mikrometruli;
5. Semasworebeli (mag. Tarazosi, ZafTa badis).
instruments aqvs limbi dayofili 360ο -ad (TviT gradusi
dayofilia minutebad), limbs aqvs Tavis damtkeci da mikrometruli
xraxnebi. limbis zemoT aris wredalidada, romelsac aqvs Tavisi damtkeci
da mikrometruli xraxni. wredalidadaze damagrebulia (amTvleli
meqanizmebi) vernierebi da cilindruli Tarazo (erTi an ori) Tavisi
Semasworebeli xraxnebiT. wredalidadaze aris agreTve ori dgari, romlis
114411
nax. 134
budeebSi Camagrebulia Wogri. erT-erT dgarze aris Sveuli wredi. Sveul
wredzec aris amTvleli meqanizmi – vernieri (SimSa-alidada) da Tarazo,
meore dgarze aris busoli.
muSaobis procesSi instrumentis nawilebis horizontalurad (an
Sveulad) dasayeneblad iaraRi aRWurvilia sami an oTxi cilindruli
TarazoTi. ori Tarazo aris horizontul wredTan (wredalidadaze) da
damagrebulia urTierT marTobulad. erTi Tarazo SimSa alidadazea
dayenebuli. meoTxe Tarazo SeiZleba damagrebuli iyos Wogrze.
instrumentis nawilebis mcire gadanacvlebisaTvis instrumenti
aRWurvilia oTxi mikrometruli xraxniT. erTi mikrometruli xraxni
mowyobilia horizontul wredTan, meore wredalidadasTan, mesame WogrTan,
meoTxe SimSa alidadasTan. TeodolitSi varCevT or RerZs: mTavar RerZs
da Wogris brunvis RerZs.
1. mTavari RerZis qveS igulisxmeba warmosaxviTi swori, romelic
gadis Wogris mzeris RerZisa da brunvis RerZis gadakveTaze,
wredalidadis da limbis centrze da xerxemala xraxnze (mas uwodeben
Sveul RerZs, rac swori ar aris), igi Sveuli maSin iqneba, rodesac limbs
moviyvanT Tarazul mdgomareobaSi.
2. Wogris brunvis RerZs uwodeben Tarazul RerZs, (rac swori ar
aris), igi Tarazuli maSinaa, rodesac limbi Tarazuladaa da dgarebi
tolia.
ZiriTadad arsebobs ori saxis
Teodoliti:
1) ganmeorebiTi; 2) ara ganme-
orebiTi.
araganmeorebiT TeodolitSi lim-
bi ar brunavs, xSuladaa wredali-
dadasTan erTad.
ara ganmeorebiTi Teodoli-
tebi 2 saxisaa.
1. limbi ar moZraobs da wreda-
lidada moZraobs an
2. roca, limbs daamagreben, wre-
114422
nax. 135
dalidada moZraobs an wredalidada da limbi damoukideblad moZraoben.
wredalidada limbs ar daetkecneba.
ganmeorebiT TeodolitSi
brunavs, rogorc limbi, ise
wredalidada. amave dros wre-
dalidada moZraobs limbTan
erTad.
vTqvaT, ganmeorebiT
TeodolitSi daketilia lim-
bi da aSvebulia wreda-
lidada, igi moZraobs. Tu
davamagrebT wredalidadas
da avuSvebT limbs, igi
miemagreba limbs da imoZ-
ravebs limbTan erTad.
13.1. xelsawyoebis Semowmebebi
Semowmebebi aris safabriko-saqarxno da savele.
Tarazo gamoiyeneba xazebis da sibrtyeebis Tarazul mdgomareobaSi
mosayvanad. mTavari anu idealuri RerZi, romlis garSemo brunavs iaraRi,
qarxanaSi winaswar dayenebulia limbis marTobulad e.i. rodesac limbis
sibrtyes moviyvanT Tarazul mdgomareobaSi, maSin mTavari RerZi dadgeba
Sveulad, radgan igi ⊥ -ia limbis sibrtyis. saWiroa limbis sibrtye
davayenoT Tarazulad.
I – Semowmeba
wredalidadaze damagrebuli Tarazos RerZi marTobi unda iyos
instrumentis mTavari RerZis.
amisaTvis ase viqceviT: iaraRs vdgamT Stativze. wred alidadaze
moTavsebul Tarazos vayenebT ori damyenebeli xraxnis gaswvriv da
xraxnebis urTierT sawinaaRmdego moZraobiT buStula mogvyavs SuaSi.
avuSvebT wredalidadis damtkecn xraxns da SemovabrunebT 180ο-iT, Tu
buStula SuaSi darCa (e.i. buStulis centri m daemTxva O wertils-
114433
nax. 136
nax. 137
nulpunqts) piroba Sesrulebulia, Tu ara gadaxris naxevars SevasworebT
Semasworebeli xraxniT, meore naxevars ki ukan gadmovaadgilebT amwevi
xraxnebiT, e.i. Tarazos 0-l punqts davamTxvevT buStulis Suaguls m-s.
isev SemovabrunebT 180ο-iT, Tu buStula SuaSi darCa, piroba Sesrulebulia,
Tu kidev gadaixara, naxevars SevasworebT Semasworebeli xraxniT, meore
naxevars ki gadavaadgilebT amwevi xraxnebiT da a.S. sanam buStula SuaSi
ar darCeba yoveli Semobrunebis dros.
I – Semowmebis geometriuli interpretacia
imis gamo, rom OK – mTavari RerZi araa marTobi H1H2 Tarazos
RerZisa da maT Soris Seiqmneba
ε -kuTxe, iZulebuli varT
xraxniT marcxena mxare avwioT,
xolo marjvena mxare davwioT ε
kuTxiT, rom buStulis Suaguli
davamTxvi-oT 0-s (nulpunqts).
axla aviRoT da wredi Semovab-
runoT 180ο-iT, e.i. ise, rom H H1 2 -
em miiRos H H1 2' ' mdeba-reoba. maT
Soris Seiqmneba β kuTxe.
β -kuTxis Sesabamisad m
gadaadgildeba m' -Si, e.i. gadaad-gilebas
Seesabameba β kuTxe an 2ε . I-el mimar-
Tulebasa da axal mimarTulebebs
Soris kuTxe naxazidan iqneba:
β ε ε ε= + − − =( ) ( )90 90 2ο ο ; (13.1.1)
β ε= 2 aqedan εβ
=2; e.i. mm'
gadaadgilebis naxevars Sevaswo-rebT
Tarazos Semasworebeli xraxniT (buS-
tulis centri m' -dan gadava m" -Si. aq vspobT erT ε -s (0 wertilTan), meore
ε -kuTxes (K wertilTan) vspobT marcxena xraxnis daweviT. amis Semdeg
Tarazos RerZi H H2 1" "⊥ gaxdeba OK (iaraRis mTavari RerZisa) da naxazi
114444
nax. 138
(211) miiRebs (212) naxazis saxes. e.i. (Tarazos H H1 2 RerZi marTobi gaxdeba
OK-iaraRis mTavari RerZis). Tarazos RerZi ⊥ -ia iaraRis RerZisa, magram
sibrtyis Tarazul mdgomareobaSi mosayvanad saWiroa ori gadamkveTi xazi
iyos Tarazulad. amisaTvis wreds SemovabrunebT 90ο -iT, Tu buStula
gadaadgilda, mas SevasworebT mesame damyenebuli xraxniT. ukve sxva
xraxnebs xels ar vaxlebT.
instrumentis qvemo nawili – limbi da wredalidada Tarazuladaa,
magram wertilebze vakvirdebiT WogriT. anaTvlebi rom swori iyos, saWiroa
instrumentis zemo da qvemo nawili geometriulad sworad iyos
dakavSirebuli. amisaTvis instruments aqvs II – Semowmeba.
Wogris mzeris RerZi marTobi unda iyos
misi (Wogris) brunvis RerZisa. (nax. 138)
Wogris mzeris RerZi aris ZafTa badis
gadakveTis da linzebis optikuri centrebis
SemaerTebeli xazi, amitom ase viqceviT:
I-li Semowmebis Semdeg 2-3 km-ze vumiznebT
raime wertils da rodesac vertikaluri wredi
marjvnivaa aviRebT βR anaTvals (saSualo orive
vernieriT) limbze. SemovabrunebT 180ο-iT,
davumiznebT imave wertils. Tu saSualo anaTvali,
rodesac wredi marcxnivaa, β L gansxvavdeba βR -sagan, maSin 180 2ο ± τ ( τ -
vernieris sizustea) marTobuloba daculia. Tu gansxvaveba metia, aviRebT
anaTvlis saSualo ariTmetikuls da wredalidadas vernieris
(mikrometruli xraxniT) davayenebT saSualo ariTmetikulze. amis Semdeg
davmzerT isev imave wertils. ZafTa badis Semasworebeli xraxnebiT
gadavwevT da vamTxvevT ZafTa badis gadakveTis samiznebel wertils.
Semdeg isev davmzerT arsebul wertils da gavimeorebT igives. Tu kidev
aRmoCnda βR da βL Soris sxvaoba ormag sizusteze meti, isev
gadavaadgilebT ZafTa bades. am moqmedebas gavimeorebT manam, sanam
kolimaciur cdomilebas ar daviyvanT minimumamde.
kolimaciuri Secdoma aris is kuTxe, romelsac Wogris mzeris
RerZi qmnis misi brunvis RerZis marTobTan. (nax. 139)
114455
nax. 139
vTqvaT, K K1 2 -s emTxveva ν ν1 2 vernierebis 0-ovani diametri da igive
K K1 2 marTobia Wogris mzeris RerZisa, maSasadame mdgomareoba
idealuria, rac ar xdeba. axla
davuSvaT, rom es piroba
darRveulia – K K1 2 -ma miiRo
axali mdebareoba, ise rom maT
Soris (K K1 2 -is marTobsa da
mzeris RerZs Soris) Seiqmna C
kuTxe. e.i. K K1 2 ar aris
marTobi mzeris RerZis. rode-
sac wredi marjvnivaa miviReT
anaTvali βR (saSualo orive
vernieridan), naxazidan
β β= −R C . (13.1.2)
axla gadavitanoT Wogri zenitze da davmziroT isev im wertils.
aviRoT (isev saSualo anaTvali orive vernierze, rodesac wredi marcxnivaa)
βL . miviRebT K K1 2 -s axal mdebareobas, romelic gadaixreba Wogris brunvis
RerZis idealuri mdebareobidan imave C kuTxiT, maSin naxazidan
β β= +L C . (13.1.3)
(2) da (3) SevkriboT da ganvsazRvroT β ; ββ β
=+R L
2; axla igive
tolobebi gadavwyvitoT C-s mimarT:
2LRC ββ −
= , (13.1.4)
maSasadame, SegviZlia ganvsazRvroT C da β (namdvili kuTxe). erTi
da igive gverds vumiznebT 2-jer. ismeba kiTxva: SeiZleba Tu ara C
gamovTvaloT miRebuli formuliT da Semdeg yvelgan SevitanoT anaTvalSi
Sesworebis saxiT? ar SeiZleba, radgan wertilebi sxvadasxva simaRlezea,
kolimaciuri Secdomis gavlenac sxvadasxva iqneba (yvela wertili, rom
erT doneze yofiliyo, gavlenac erTi da igive iqneboda). amitom
mimarTulebis gansazRvrisas kolimaciuri cdomilebis gansazRvra da
Semdeg Sesworebis saxiT misi Setana ar SeiZleba. aucileblad saWiroa
yoveli damzeris dros anaTvlis aReba vertikaluri wredis or
114466
nax. 140
mdgomareobaSi β R da β L . maTi saSualo ariTmetikuli mTlianad
gamoricxavs minimumamde dayvanil kolimaciur cdomilebas. Tanamedrove
instrumentebSi ZafTa badis gadasawevi xraxnebi naklebadaa xel Sesaxebi.
amas uzrunvelyofs qarxana.
III – Semowmeba
Wogris brunvis RerZi marTobuli unda iyos iaraRis mTavari RerZisa.
e.i. dgamebi unda iyvnen urTierTSoris toli da paralelurebi. (nax. 140)
gavalT Senobidan ≈ Senobis si-
maRlis tol manZilze (vTqvaT 30m-ze) da
davumiznebT maRal wertils. Semdeg
nelnela davuSvebT Wogrs Cvens simaR-
leze (ise, rom SevZloT damzera)
zustad. zemo wertils davaniSninebT
ZafTa badis gadakveTis wertilze.
Semdeg Wogrs gadavitanT zenitze,
SemovabrunebT 180ο-iT da isev gavi-
meorebT zemod aRweril moqmedebas. Tu
ZafTa badis gadakveTa daemTxva Cvens
daniSnul wertils, dgamebi Tanabari
yofila, magram Tu ar daemTxva, Secdomas mainc vspobT orjer damiznebis
Semdeg.
vTqvaT, davumizneT T-wertils. davuSviT Wogri da aRvniSneT A-
wertili, Semdeg SemovabruneT wredi 180ο-iT, Wogri gadavitaneT zenitze da
isev gavimeoreT, axla gamosaxulebas A-s nacvlad miviRebT B-Si, maSin AB
xazs gavyofT Suaze da davniSnavT C wertils. Cven ki Wogrs
gadavaadgilebT wredalidadas mikrometruli xraxniT C-saken. axla isev
davmzerT T-s, magram igi gamoCndeba ara T-wertilSi, aramed T' -Si, e.i.
dgamebi toli araa, rom gavaTanabroT saWiroa erTi romelime dgamis
simaRlis Secvla manam, sanam ZafTa badis centri ar daemTxveva T-wertils.
es moqmedeba sruldeba dgamis Semasworebeli xraxnebiT.
igive Semowmeba mSvidi amindis pirobebSi SeiZleba gavakeToT Sveulis
gamoyenebiT (nax. 141).
114477
nax. 141
nax. 142
davkidebT Sveuls da davmzerT, Tu Sveuli gayva ZafTa badis Sveul
Zafs, piroba Sesrulebulia,
Tu gadaixara marcxniv
(miiRo 1-li mdebareoba),
marcxena dgami yofila
mokle, Tu gadaixara mar-
jvniv (miiRo 2-re mdeba-
reoba), marjvena dgami yo-
fila mokle, Tu miiRo 3-me
mdgomareoba, maSin 1-li
Semowmeba ar aris swori.
wredalidadas eqscentrisiteti (nax. 142)
aviRoT C centri da nulovani diametri 0 180ο ο− . vTqvaT limbisa da
wredalidadas centrebi emTxveva erT-
maneTs. SemovabrunoT wredalidada,
aviRoT anaTvali β , romlis Sesabamisi
centraluri kuTxea β . igi izomeba
Sesabamisi rkaliT. axla davuSvaT,
rom qarxana Secda da wredalidadas
centri ar emTxveva limbis centrs.
misi centri C-s nacvlad C1-Sia,
maSasadame alidada Semobrundeba C1-
is garSemo. igive kuTxuri gadax-
risaTvis β -s nacvlad anaTvali iqneba
β1 . gansxvaveba β β− =1 X (xazovani
gansxvaveba). X-is Sesabamisi kuTxuri
gansxvaveba iqneba ε . C1C-s ewodeba
eqscentrisiteti. C1-dan davuSvaT marTobi ν ν1 2 diametrze, miviRebT C2-
wertils.
X C C r= = ⋅ = ⋅1 2 sin sinε βλ , (13.1.5)
sadac ε aris (X-xazovani Secdomis Sesabami kuTxuri Secdoma) mcire kuTxe
da SegviZlia SevcvaloT Sesabamisi radianuli zomiT:
114488
r ερ
β""
sin= ⋅λ , maSasadame
ε ρ β" " sin= ⋅ ⋅λr
, (13.1.6)
λr ewodeba fardobiTi eqscentrisiteti (mudmivi sididea). ρ aris radiani
= = = =57 3 3438 206265 11
ο, ' "sin "
(mudmivia) e.i. ε" -is cvalebadoba damokidebulia
β -ze, naxazidan β β= +1 X ,
β β= − −2 180X ο .
es ori gantoleba gadavwyvitoT β -s mimarT, ββ β
=+ −1 2 180
2( )ο
.
X-cvalebadi sididea, igi sxvadasxva mdebareobis dros icvleba max-dan
min-mde 0-mde. (mag. rodesac β = 0οan 180ο , maSin X = 0 (anu min) da rodesac
β = 90οan β = 270ο , am dros X = λ anu max-s), amitom orjer anaTvlis aRebas
azri aqvs. e.i. eqscentrisitetis mosaspobad vsargeblobT orive vernieriT. I-
vernieriT viRebT gradusebs da minutebs, II-vernieriT mxolod minutebs.
13.2. Sveuli wredis nul adgili
daxris kuTxe ewodeba mocemul mimarTulebasa da Tarazul
mimarTulebas Soris Seqmnil Sveul kuTxes.
Cven viciT, rom iaraRSi gvaqvs Sveuli wredi, romelic daxris kuTxis
gansazRvrisaTvis gamoiyeneba. daxris kuTxe ki gvWirdeba wertilebs Soris
aRmatebis gansazRvrisaTvis. Sveuli wredi mimagrebulia dgarze uZravad.
Cven vmzerT WogriT. Tu Sveul wredsa da Wogris mzeris RerZs Soris ar
iqna swori geometriuli damokidebuleba, anaTvali ar iqneba swori.
daxris kuTxe, rom ganisazRvros, saWiroa Tarazuli mimarTuleba.
Tarazul mimarTulebas gvaZlevs Tarazo.
iaraRs aqvs Wogri – mzeris RerZiT, Sveuli wredi nulovani diametriT,
Tarazo – Tavisi RerZiT da vernierebi – vernierebis 0-ebis SemaerTebeli
xaziT. maSasadame, ZiriTadad saqme gvaqvs 4 RerZTan.
114499
1. Wogris mzeris RerZi da Sveuli wredis 0-vani diametri unda iyvnen
paralelurebi
2. Sveul wredze Tarazos RerZi da vernierebis diametri unda iyos
paraleluri. e.i. Tu wyvil-wyvilad es RerZebi paralelurebi arian,
anaTvali iqneba 0. Tu romelime piroba darRveulia, Sveul wredze
adgili eqneba nul adgils, rac uaryofiTi movlenaa instrumentSi.
Sveuli wredis nul adgili aris anaTvali Sveul wredze maSin,
rodesac Wogris mzeris RerZi da SimSa alidadaze damagrebuli Tarazos
RerZi arian Tarazulad.
normalur instrumentSi igi unda iyos 0-is toli. Tu is 0-s ar
udris amis mizezi iqneba, erTis mxriv Wogris mzeris RerZisa da Sveuli
wredis 0-ovani diametris araparaleluroba.
ganvixiloT arsebuli Sveuli wredebis ZiriTadi tipebi
nax. 143
I-aris limbis tipis. sawyisi aris erT adgilze da bolo
danayofebi izrdeba 360ο -mde.
II-Si gamoyenebulia orive bolo. iwyeba bolodan, grZeldeba 360ο -
mde. davuSvaT, rom Wogris mzeris RerZi Tarazuladaa da paraleluria
Tarazos H1H2-RerZisa. am SemTxvevaSi Tu υ υ1 2 -vernierebis 0-ovani diametri
emTxveva am RerZebs, maSin anaTvali iqneba 0-li. axla davuSvaT rom H1H2
ar emTxveva υ υ1 2 -s. gvaqvs Sveuli wredi marjvniv da vmzerT raRac B
wertils. mzeris RerZsa da nulovan diametrs Soris Seiqmneba raRac ε1
kuTxe. meore mxriv υ υ1 2 da H H1 2 -s Soris ε 2 -kuTxe, namdvili daxris kuTxea-
ν , romelSic Seitaneba Secdoma ε1 da ε 2 .
115500
nax. 144
nax. 145
ε ε1 2+ -aRvniSnoT – N da vuwodoT nul adgili (Secdoma) e.i.
SecdomiT aRebulia kuTxe νR .
ν ν ε ε ν= − + = −R R N( )1 2 . (13.2.1)
axla gadavitanoT Wogri
zenitze da davayenoT wredi mar-
cxniv. Cven gvainteresebs isev ν
kuTxe (nax. 146)
ν ν ν= − + = −360οL LN N . (13.2.2)
νν ν
=−R L
2. (13.2.3)
N R L=+ν ν2
. (13.2.4)
moviyvanoT magaliTi
νR = 9 40ο ' ;
νL = 350 10ο ' ,
marjvena anaTvals meqanikurad
vumatebT 360ο , maSin
N =+ +
=( ' ) ' '9 40 360 350 10
2359 55
ο ο οο .
kontrolis mizniT gamovi-
yenoT (13.2.2) formula
ν = − = +359 55 350 10 9 45ο ο ο' ' ' ,
gansxvaveba miviReT (+) niSniT e.i.
yofila aRmarTi.
gamoviyenoT (13.2.1)-i formula
ν = − = +369 40 359 55 9 45ο ο ο' ' ' .
gamoviyenoT (13.2.3)-me formula ν =−
= +369 40 350 10
29 45
ο οο' ' '
es n. a. didia Cveni anaTvali '409ο=Rν
n. a. = 5' da ν kuTxe 9 40ο ' nacvlad gamovida 9 45ο ' .
nuladgilis minimumamde dayvana xdeba ori xerxiT.
115511
1. vTqvaT, gvaqvs 9 45ο ' , romelic miviReT gamoyvaniT, anaTvali νR = 9 40ο ' ,
vertikaluri wredis mikrometruli xraxniT Sveuli wredis vernierebis
nulovani diametri davayenoT 9 45ο -ze, axla Tarazos buStula gadaixreba,
Tarazos Semasworebeli xraxnebiT Tarazos buStulis gadaxras
SevasworebT.
2 xerxi. buStulas moviyvanT SuaSi, Semdeg Wogrs daaxloebiT
davayenebT Tarazulad da anaTvals davayenebT 359 55ο ' (e.i. im anaTvalze,
romelic miviReT gamoTvliT), amis Semdeg vertikaluri wredis
mikrometruli xraxniT anaTvals gadavaadgilebT 0-mde, maSin vertikaluri
wredis Tarazos buStula gadaixreba. axla buStulas moviyvanT SuaSi
Tarazos Semasworebeli xraxnebiT.
kolimaciuri Secdoma mudmivia, gavlena cvalebadi, asevea
eqscentrisiteti. magram aq n. a. SeiZleba erTxel ganisazRvros da Semdeg
SevitanoT Sesworebis saxiT.
Tu davakvirdebiT naxazs, rodesac Sveuli wredi marjvniv gvaqvs,
anaTvals viRebdiT jer I-li, Semdeg II-re vernieriT. axla vTqvaT gvaqvs
wredi marcxniv da anaTvals viRebT jer II-e, xolo Semdeg I-li vernieriT,
maSin (13.2.1) formula ZalaSi rCeba, (13.2.2)-is nacvlad gveqneba (13.2.2' ).
ν ν= − +180οL N ; (13.2.2' )
ν ν ν= − +R L N' ; (13.2.3' )
N R L= + ±ν ν' 180ο . (13.2.4' )
e.i. I-li tipis wredisaTvis (pirvel SemTxvevaSi), formulaSi emateba
360ο , rodesac vsargeblobT I-lis Semdeg II vernieriT da gvaqvs Sveuli
wredi marjvniv, xolo rodesac Sveuli wredi marcxnivaa, anaTvals
pirvelad aviRebT II-vernieriT, Semdeg I-iT da formulaSi vamatebT 180ο-s.
I-li tipis dros vsargeblobT pirveli 4-formuliT, xolo II-re
tipisaTvis 1-li ZalaSi rCeba da gamoiyeneba agreTve 2 3 4' ' ' .
III-tipisaTvis (1)-li isev ZalaSi rCeba, danarCeni formulebi ki miiRebs
aseT saxes:
ν ν= +L N ; (13.2.2" )
νν ν
=+R L
2; (13.23" )
115522
nax. 146
nax. 147
N R L=−ν ν2
, (13.2.4" )
magram niSani Cven unda davimaxsovroT. velze niSnis dasma, Tu
rogoria aRmarTi, Tu daRmarTi, xandaxan Zneldeba, amitom am wredebSi
TviTon formula gvaZlevs niSans.
Tu yovelTvis anaT-vals viRebT im vernieriT, romelic Cvenskenaa,
maSin vsargeblobT I-li
formuliT. II tipis
wreds Seesabameba pir-
veli 4 formula, mxo-
lod mcires vamatebT
360ο -s.
Sveuli wredis
nuladgili roca
Wogrze damagrebulia
Tarazo
am SemTxvevaSi I-
el rigSi unda davr-
wmundeT, Tarazos RerZi
aris Tu ara para-
leluri Wogris mzeris
RerZisa da Semdeg mov-
spoT nul adgili, Tu
igi arsebobs. amisaTvis
SevarCevT daxril ad-
gils da vasrulebT
ormag nivelobas. A da
B wertilebs Soris simaRleTa sxvaoba vTqvaT aris h. A wertilSi
davcentreT Teodoliti, B wertilSi ki davayeneT lartya. RerZebi rom
erTmaneTs emTxveodnen, anaTvali iqneboda swori, magram radgan RerZebi ar
emTxvevian, arsebul a1 anaTvalSi adgili eqneba Δa cdomilebas gamowveuls
115533
Tarazos RerZisa da mzeris RerZis ara paralelobiT. Cven gvainteresebs
arsebobs Δa Tu ara. naxazidan:
h i a a= − +1 1( )Δ . (13.2.5)
axla SevabrunoT, instrumenti davayenoT B – wertilSi, A – wertilSi
ki lartya. am SemTxvevisaTvis simaRleTa sxvaoba
h a a i= + −( )2 2Δ . (13.2.6)
gadavwyvitoT orive gantoleba Δa -s mimarT
Δaa a i i
=+
−+1 2 1 2
2 2, (13.2.7)
e.i. cdomileba ar aris 0-is toli. anaTval a2-s davamatebT Δa -s, an Wogrs
gadavaadgilebT Δa -s mimarT, Tarazos buStula gadaiwevs. Tarazos
SevasworebT Semasworebeli xraxniT. aviRoT sxvaoba a a2 − Δ da am
anaTvals lartyaze davumiznoT Wogris mzeris RerZi. masze mowyobili
Tarazos buStula am dros gadaixreba. es gadaxra SevasworoT Tarazos
Semasworebeli xraxnebiT. am moqmedebiT Tarazos RerZi Wogris mzeris
RerZis paraleluri gaxda da orive Tarazulia. Semdeg daviwyoT nul
adgilis Sesworeba. SimSa alidadas mikrometruli xraxnis saSualebiT
davayenoT Sveul wredze anaTvali nulze. Tu SimSa alidadaze mdebare
Tarazos buStula isev gadaixara, moviyvanoT igi nul punqtSi, am Tarazos
Semasworebeli xraxnebiT.
13.3. instrumentis dacentvrisa da reduqciis Secdoma
1. dacentvris Secdoma. kuTxe, rom gavzomoT, instrumenti unda
davayenoT kuTxis wveroSi. vTqvaT, limbis centri zustad daemTxva C-
wertils. Cveni mizania β kuTxis gazomva. Tu gvaqvs limbi 0-ovani
diametriT, avuSvebT wred alidadas da davumiznebT A – wertils, miviRebT
βA -polarul kuTxes, aseve aviTvliT βB -s. miRebuli anaTvlebiT
ganvsazRvravT β . < = −β β βB A , Tu marjvena anaTvali mcire iqneba marcxenaze,
vamatebT 360ο -s. miRebuli kuTxis mniSvneloba swori iqneba, Tu instrumenti
zustad C wertilSi davcentreT, magram davuSvaT rom sxvagan davcentreT
115544
nax. 148
nebiT Tu uneblieT, vTqvaT I-wertilSi. maSasadame, wredi gadmoinacvlebs,
orientacia igive aviRoT (nax. 148).
manZils CI-aRniSnaven λ-iT da uwodeben dacentvris xazovan elements
e.i. CA DA= -s nacvlad
gveqneba ADIA '= .
Q-kuTxe aris dacen-
tvris kuTxuri elementi.
BDCB = nacvlad gveq-neba
BDIB '= .
C1 aris dacentvris
kuTxuri Secdoma, roca
vmzerT A-s, C2 aris
dacentvris kuTxuri Sec-
doma, roca vmzerT B. Cven
gvWirdeba βA naxazidan
β ββ β
A A
B B
CC
= += +
' ,' .
1
2
(*)
nacvlad β kuTxisa, vsazRvravT M kuTxes, e.i. unda SevitanoT
Secdoma C1 da C2. (*)-dan ganvsazRvroT β
β β= − + − = +' ' ( )B AB C C M C2 1 , (13.3.1)
sadac C C C2 1− = (aRvniSneT), axla ganvsazRvroT cal-calke C1 da C2.
naxazze sinusebis Teoremis safuZvelze, CIAΔ -dan QD
CA
sin"1 ⋅⋅= ρλ aseve
)sin("2 MQD
CB
+⋅⋅= ρλ. sadac, rogorc zemoT aRvniSneT, λ-dacentvris
xazovani elementia, ρ -radiani= 206265" . formulaSi cvalebadi sididea DA-
manZili. igi ukuproporciul damokidebulebaSia gazomvis sizustesTan. e.i.
gazomvis dros velze xazebi unda SevarCioT rac SeiZleba grZeli, maSin
dacentvris Secdoma is gavlena iqneba mcire.
115555
nax. 149
2. reduqciis
Secdoma. davuSvaT,
rom instrumenti zus-
tad davcentreT da
< = −β β βB A , magram
zustad ver davmzi-
reT. adgili aqvs wina-
aRmdegobas. davmzireT
A wertili da gamo-
saxuleba miviReT A' -
Si_ D A' ; <r-ewodeba
reduqciis kuTxuri Secdoma, rodesac vmzerT A wertils. λA -ewodeba
reduqciis Secdoma, rodesac vmzerT A wertils. aseve reduqciis gamo, B-e
davmzireT B' -Si da analogiurad miviReT BD' , λB da r2 . e.i. Cven β -s
nacvlad ganvsazRvreT M.
A – wertilisaTvis β βA A r= −' 1 ,
B – wertilisaTvis β βB B r= +' 2 ,
orivedan
β β β= − + − = +' ' ( )B A r r M r2 1 . (13.3.2)
ΔCAA' -dan rD
QA
AA" sin1 = ⋅ ⋅
λρ ; ΔCBB' -dan r
DQB
BB" sin1 = ⋅ ⋅
λρ .
aqac, ise, rogorc dacentvris dros mniSvnelSi manZilebia, e.i.
reduqciis gavlena rom naklebi iyos, saWiroa gverdebi iyos grZeli.
13.4. kuTxeebis gazomva
magnituri azimutis gansazRvris Semdeg, busols vxsniT da viwyebT
kuTxeebis gazomvas. imis gamo, rom kuTxeebis gazomva Tanxvdenilia
SemTxveviTi da sistematuri cdomilebebiT (mag. vTqvaT romelime 359
anaTvali ar aris sworad dakvesili, maSin danarCeni anaTvlebi Tanxvdenili
115566
nax. 150
nax. 151
iqneba sistematuri cdomilebiT, an vTqvaT, romelime kvesuri msxviladaa
dakvesili, an ar gadis zustad centrSi, adgili eqneba mcdar anaTvlebs).
arsebobs kuTxeebis gazomvis sxvadasxva xerxebi: ganmeorebiTi, ileTebis,
wriuli ileTebis, alajalovis, Sreiberis da sxva mravali. (SeviswavloT
ileTebis, wriuli ileTebis da ganmeorebiTi xerxebi).
a) ileTebis xerxebi
ileTebis xerxiT kuTxeebis gazomvis
dros erTi da igive mimarTulebas vumiznebT
Wogrs ramodenimejer, limbis sxvadasxva
nawilebis gamoyenebiT da yoveli damiz-
nebisas limbze viRebT anaTvlebs. vsazRvravT
yoveli mimarTulebis ualbaTes mniSvnelobas
(saSualo ariTmetikuls) da bolos gani-
sazRvreba kuTxe. ileTebis xerxiT kuTxis
gasazomad gamoiyeneba formula _ 180 1
ο
ni( )− ,
sadac n–ileTebis ricxvia, i–ileTebis
Tanmimdevroba. vTqvaT n=2 niSnavs,
gamoviyenebT wredis or nawils, i = 1 e.i.
pirveli ileTi 180
21 1 0
οο( )− = , roca i = 2 e.i.
meore ileTi 180
22 1 90
οο( )− = ;
axla vTqvaT, kuTxes vzomavT 3 ileTiT
I-ileTi
1803
1 1 0 0 60
1803
2 1 60 0 120
1803
3 1 120 0 180
οο ο ο
οο ο ο
οο ο ο
( )
( )
( )
− = −
− = −
− = −
6
12
maSasadame, limbi daiyofa Semdegnairad:
115577
nax. 152
vTqvaT, unda gavzomoT ACB = β kuTxe, Tu ganmeorebiTi Teodoliti
gvaqvs SevaTavsebT 0-s 0-Tan (Tu araa ganmeorebiTi Teodoliti, maSin 0 0ο ο-
ar SeTavsdeba). sinamdvileSi 0ο -ki ar vaTavsebT, gadavacilebT raRac 5' -ze
(nax. 152) anaTvlebis ukeT aRebis mizniT. (vumzerT saris ZirSi, zemoT ara,
igi SeiZleba gadaxrili
iyos). gvaqvs wredi marjvniv,
RA-anaTvali gveqneba A-
wertilze damiznebisas. lim-
bi damagrebulia, avuSvebT
alidadas da davmzerT B
wertils, aviRebT anaTvals
RB-ze (Sesrulda I-ileTi).
gadavitanT Wogrs zenitze da davmzerT isev A-s (am SemTxvevaSi wredi
marcxnivaa). limbi damagrebulia, wredalidada moZraobs. analogiurad
gveqneba LA da LB anaTvlebi. mimarTuleba A wertilze imzireba orjer da
aiReba saSualo anaTvali
βAA AR L
=+2
; (13.4.1)
analogiurad βBB BR L
=+2
; (13.4.2)
kuTxe β β β= − =+ − +
B AB B A AR L R L[ ] [ ]
2, (13.4.3)
maSasadame, erTi β kuTxis gasazomad yovel mimarTulebas davmzireT or-
orjer. aviReT anaTvali 4-jer. e.i. erTi ileTiT gazomvisas erT
mimarTulebaze viRebT ori anaTvlis saSualo ariTmetikuls da meore
mimarTulebazedac saSualo ariTmetikuls.
ori ileTiT gazomva niSnavs gamoviyenoT wredis 2-nawili 0 90ο ο− -mde
da am nawilis simetriuli 180 270ο ο− -mde. da gazomvis dros igive
moqmedebebs gavimeorebT.
sami ileTis SemTxvevaSi 90ο -is nacvlad gveqneba 60 120 180− −o o o
maSasadame, n-ileTis SemTxvevaSi formula miiRebs saxes.
β β βn Bn AnB B
nA A
nR L R Ln
= − =+ − +
⋅[ ] [ ]1 1
2, (13.4.4)
sadac n-ileTebis raodenobaa,
115588
nax. 153
RA da RB –anaTvlebi limbze, rodesac Sveuli wredi marjvnivaa;
LA da LB – anaTvlebi limbze, rodesac Sveuli wredi marcxnivaa.
β _ kuTxis n-ileTiT gansazRvrisaTvis limbze anaTvlebs viRebT
4n-jer, aseve A da B sagnebs WogriT vumiznebT 4n-jer.
b) wriuli ileTebis xerxi
es xerxi gamoiyeneba, rodesac saqme gvaqvs ramdenime mimarTulebasTan,
Cveni mizania ganisazRvros 5 kuTxe β β β β β1 2 3 4 5, , , , naxazidan
β β β1 = −B A ,
β β β2 = −C B ,
β β β3 = −D C ,
β β β4 = −E D ,
β β β5 360= + −( )A Eο .
gansxvaveba ileTebsa da wriul
ileTebs Soris aris is, rom wriul
ileTebSi erTdroulad izomeba ramdenime
kuTxe da pirveli anaTvali iReba orjer
kontrolis mizniT.
muSaobis TanmimdevrobaSi formula
rCeba (ileTisaTvis), magram wriul ileTebSi daculi unda iqnas
Tanmimdevroba: vTqvaT davumizneT A wertils, aviReT anaTvali RA, Semdeg RB,
RC da a.S. isev R A' -s kontrolis mizniT saaTis isris mimarTulebiT. Tu
iaraRi mdgradia, RA unda daemTxvas R A' -s. vTqvaT ar daemTxva, maSin maT
Soris sxvaoba ar unda aRematebodes instrumentis sizustes, e.i.
R RA A− ≤' τ , sadac τ -instrumentis sizuste. alidadis marjvniv brunviT da
yvela mimarTulebaze anaTvlebis aRebiT mTavrdeba naxevari wriuli ileTi.
Semdeg gadavitanT Wogrs zenitze da alidadas brunviT vumzerT sawyis A-
punqts. viRebT anaTvals, Semdeg alidadas marcxniv brunviT Sebrunebuli
TanmimdevrobiT damzeren punqtebs. analogiurad gveqneba anaTvlebi LA, LF,
115599
nax. 154
LC, LD, LB da L'A , aqac L L'A A− ≤ τ (mowodebulia gausis mier). amiT
damTavrda erTi wriuli ileTi.
maSasadame wriul da Cveulebriv ileTebs Soris is gansxvavebaa, rom
yovel naxevar ileTSi sawyis wertils vumzerT orjer, vamowmebT
instrumentis mdgradobas.
g) ganmeorebiTi xerxi
ganmeorebiTi xerxiT kuTxis gazomvisaTvis aucilebelia ganmeorebiTi
Teodoliti. am xerxis arsi SemdegSi mdgomareobs:
erTi da igive mimarTulebas vumiznebT Wogrs ramodenimejer,
risTvisac viyenebT wredis sxvadasxva nawilebs, xolo anaTvlebs limbze
viRebT oTxjer (geodezistebi) an orjer (markSeiderebi). am xerxiT iseve,
rogorc ileTebis (wriuli ara) xerxiT, izomeba mxolod erTi kuTxe, or
mimarTulebas Soris. praqtikulad, yovel wyvil damiznebisas vzomavT
kuTxeebs da viRebT maT saSualo ariTmetikuls.
ganvixiloT magaliTi: vTqvaT unda gavzomoT β kuTxe (ix. nax. 154)
SevuTavsebT limbis 0ο–s alidadis 0ο -s, ufro sworad cotaTi gadavacdenT
5' an 10' , davumiznebT A
wertils Teodolitis Sve-
uli wredis marjvniv
mdgomareobaSi. I damizneba
mogvcems RA (1) anaTvals,
limbi damagrebulia. avuS-
vebT wredalidadas da
mivmarTavT B-saken, miviRebT
RB (2) anaTvals (aq aucilebelia ganmeorebiTi Teodoliti, rom wredali-
dada miemagros limbs). axla davamagrebT wredalidadas limbs da
avuSvebT limbs. mivmarTavT A-saken, Tan Wogrs gadavitanT zenitze.
maSasadame (1) da (2) anaTvlebis Sesabamisi seqtori < βR gadainacvlebs
(naxazze zemoT) βR B AR R= − ; βL B AL L= − ; davmzerT isev A wertils, anu
aviRebT anaTvals (3) LA analogiurad. wredalidadis moZraobiT aviRebT
anaTvals LB (4). LA da LB-saSualebiT miviReT βL mniSvnelobas. maSasadame β
116600
nax. 155
kuTxe gavzomeT 2-jer, an anaTvali aviRebT 4-jer, e.i. erTi da igive kuTxe
gaizoma 2-jer da aviReT saSualo ariTmetikuli.
ββ β
=+⋅
=− + −
⋅R L B A B A
nR R L L
n2 2( ) ( )
, (13.4.5)
Tu davakvirdebiT formulas ileTebisaTvis da ganmeorebisaTvis,
naTlad davinaxavT, rom maT Soris araviTari gansxvaveba ar aris.
erTi ganmeoreba da erTi ileTi erTi da igivea. n ganmeorebiT
gazomvis dros dakvirvebas imeoreben n-jer, yovelTvis limbis moZraobiT
marcxena saganze da alidadis moZraobiT marjvena saganze (damiznebiT).
alidada da limbi brunavs erTi mimarTulebiT ukanasknelad kuTxis n-jer
gazomvis Semdeg, roca alidada dayenebulia marjvena B-saganze aiReba
anaTvali. maSasadame gasazomi kuTxis limbze n-jer gazomvisas anaTvali
aiReba mxolod dasawyisSi da boloSi.
anaTvlebis mcire raodenobis saWiroeba warmoSva anaTvlis SecdomiT
aRebam. e.i. rac metjer viRebT anaTvals, miT met Secdomebs vuSvebT.
axla vnaxoT, rogor izomeba kuTxe miwis qveS. markSeiderebi muSaoben
miwis qveS Znel pirobebSi. maTi sizuste dedamiwis zedapirze geodeziuri
samuSaoebis sizustes emyareba. amitom miwis qveS mzeren orjer. aqac
saWiroa ganmeorebiTi
Teodoliti. vaTavsebT 0-
s 0-Tan. davmzerT A-s da
aviRebT anaTvals RA–s,
vTqvaT 15' -ia. davamagrebT
limbs, avuSvebT
wredalidadas da
davmzerT B-s, anaTvals
ar viRebT. davamagrebT wredalidadas, avuSvebT limbs, Wogrs gadavitanT
zenitze da davmzerT isev A-s. aqac seqtori gadainacvlebs da anaTvlebs
aviRebT bolos
β =−L RB A
2, (13.4.6)
es aris erTi sruli ganmeoreba. ramdeni ganmeorebac ar unda iyos, anaT-
vals viRebT orjer. ori ileTiT gazomvis dros vmzerT 8-jer. anaTvals
116611
viRebT 2-jer. RA da RB sxvaobas vyofT 2n-ze. yovelTvis limbi moZraobs
marcxniv, wredalidada marjvniv, n-ganmeorebaTa ricxvia.
ganvixiloT kuTxeebis Caweris rogori forma arsebobs (vTqvaT
wriuli ileTebis SemTxvevisaTvis):
1-2-3 grafa rom Seivseba, gadavitanT Wogrs zenitze da isev aviRebT
anaTvals SevavsebT 4 grafas. gansxvaveba gradusebSi unda iyos 180ο , amis
Semdeg gamoiTvleba danarCeni grafebi.
mag. 3 da 4 svetis saSualo anaTvali Caiwereba 5-Si.
6-svetSi Caiwereba ormagi kolimaciuri Secdoma, romelic daiyvaneba
minimumamde ZafTa badis gadaweviT.
anaTvali
wredze sadguri
# _ #
wertili
# R↓ L↑
saS. LR LRC −=2
2LR + mim.
1 2 3 4 5 6 7 8
0
A
B
C
D
E
A
'16'180ο
'20'2242ο
'24'2676ο
'28'30182ο
'32'34275ο
'17'190ο
'16'18
'20'22
'28'26
'32'36
'17'190ο
'24'2142ο
'25'2776ο
'29'31180ο
'33'35275ο
'18'190ο
_ 2
_ 2
_ 2
_ 2
_ 2
_ 1
116622
nax. 156
13.5. gazomili kuTxeebis gawonasworeba
rac ufro metjer iqneba gazomili kuTxe, miT ufro zusti iqneba
saSualo ariTmetikuli, magram kuTxeebis absoluturi mniSvneloba mainc
ar gveqneba. amitom saWiroa kuTxeTa ualbaTesi sididis dadgena e.i.
cdomilebis gawonasworeba.
vTqvaT, mocemulia elementaruli
figura Δ . Δ –Si α , β , γ -kuTxeebi. α -s, β -s
da γ -s miRebisaTvis gazomili gvaqvs
damoukideblad kuTxeebi da calkeuli
gazomvebiT miviReT kuTxeebi α , β , γ .
α -Tvis . . . . . . . α1 , α 2 , α3 , . . . . . . . α n ,
aseve β -Tvis . . . . . . . β1 , β2 , β3 , . . . . . . . β n ,
γ -Tvis . . . . . . . γ 1 , γ 2 , γ3, . . . . . . . γ n .
cxadia samkuTxedis yoveli kuTxis ualbaTesi sidideebi
ganisazRvreba
αα
=n
; ββ
=n
; γγ
=n
, (13.5.1)
e.i. α , β , γ - ualbaTesi sidideebia. miuxedavad amdeni gazomvebis
warmoebisa, mainc α β γ+ + ≠ 180ο e.i. Teoriuls ar udris, maSasadame
α β γ+ + − =180ο W , (13.5.2)
adgili aqvs Seukvrelobas - W, romelic aris Sedegi am sami kuTxis
gazomviT miRebuli, magram, kerZod, romeli kuTxis gazomvaSi davuSviT
Secdoma, ar viciT. imis gamo, rom gazomva xdeba erTi instrumentiT, erTi
adamianis mier, erTi da igive pirobebSi, gazomva aris tolzusti. W-s ki
gaawonasworeben ase: ( ) ( ) ( )α ε β ε γ ε+ + + + + − =1 2 3 180 0ο e.i. Sesworebis Setanis
Semdeg unda gautoldes Teoriuls an Sesworebulisa da Teoriulis
sxvaoba unda gaxdes 0-is toli. Secdoma udris, Sesworebebs Sebrunebuli
niSniT, magram maT erTmaneTTan zogjer kavSiri ara aqvT.
ε ε ε1 2 3+ + = W an ε ε ε1 2 3 0+ + − =W ; sadac W= + + −α β γ 180ο (13.5.3)
116633
nax. 157
nax. 158
es aris saerTod figuris piroba, mocemuli SemTxvevisaTvis ki
samkuTxedis piroba an Sesworebis piroba. es meTodi mowodebulia gausis
mier.
samkuTxedSi an mravalkuTxedSi Siga kuTxeebis jami unda gautoldes
Teoriuls
2 2d n( )− . (13.5.4)
analogiurad poligonis piroba
iqneba
ε ε ε1 2 6 2 2 0+ + + − − + =... ( )d n W , (13.5.5)
W d n= + + + − −β β β1 2 6 2 2... ( ) . (13.5.6)
esaa Sekruli poligonis piroba.
13.6. direqciuli kuTxeebis gamoTvla
Cven ganvixileT sawyisi mimarTulebis azimutis gansazRvra, ris
Semdeg instruments vxsniT yiblans, xolo Semdeg cnobili direqciuli
kuTxiTa da
gawonasworebuli
kuTxeebiT vsazRvravT
danarCeni gverdebis
direqciul kuTxeebs.
vTqvaT, mocemulia
α1 2− , viciT α δ= +Agο ; α -
direqciuli kuTxea, Ag -
geografiuli azimuti, δο-
mixriloba. Cveni mizania
ganvsazRvroT α 2 3− direq-
ciuli kuTxe naxazidan
α α β2 3 1 2 2180− −= + −ο .
116644
yoveli gverdis direqciuli kuTxe udris, wina gverdis direqciul
kuTxes (+) 180ο-i da (_) am gverdebs Soris marjvena kuTxe. axla vTqvaT
gvaqvs ara β2 aramed β'2 -marcxena kuTxe, naxazidan
2 3 1 2 2 1 2 2180 (360 ) 180α α β α β− − −= + − − = − +o o o .
yoveli gverdis direqciuli kuTxe udris, wina gverdis direqciul
kuTxes (-) 180ο-i da (+) am gverdebs Soris svlis marcxena kuTxe. Tu
saklebs maklebi ar akldeba, vamatebT 180ο . axla gamoviyvanoT wesi
zogadad.
yoveli gverdis direqciuli kuTxe udris, wina gverdis direqciul
kuTxes (+) an (_) 180ο da minus marjvena kuTxe am gverdebs Soris an plius
marcxena kuTxe (es damokidebulia saiT gveqneba svla). kontroli aris
gamoTvla imave gverdis direqciuli kuTxisa, romeli gverdidanac daviwyeT.
13.7. pirdapiri da Sebrunebuli geodeziuri amocana
geodeziuri amocanebi orgvaria: 1) pirdapiri da 2) Sebrunebuli
pirdapiri ewodeba amocanas, rodesac mocemulia erTi sawyisi
wertilis koordinatebi, manZili meore wertilamde da mimarTuleba anu
direqciuli kuTxe da unda ganisazRvros meore wertilis koordinatebi.
(moc. sawyisi wertilis koordinatebi ar unda iyos zoldneris an gausis).
mocemuli polaruli koordinatebiT veZebT marTkuTxa koordinatebs.
Sebrunebuli geodeziuri amocana mdgomareobs SemdegSi: mocemuli ori
wertilis koordinatebiT unda ganisazRvros mimarTuleba anu direqciuli
kuTxe da manZili mocemul or wertils Soris. e.i. mocemuli marTkuTxa
koordinatebiT veZebT polarul koordinatebs (mocemul 2 wertilis
koordinatebs mniSvneloba ara aqvs raze iqneba, sibrtyeze, sferoidze Tu
sferoze). TviT koordinatebis gansazRvra sferoze da sferoidze, miT
umetes, Tu saqme gvaqvs geografiul koordinatebTan (rasac saerTaSoriso
mniSvneloba aqvs) Zalian Znelia.
116655
nax. 159
pirdapiri da Sebrunebuli geodeziuri amocanebis gadawyveta xdeba
sferoidis zedapirzedac, rac ufro rTulia. ganvixiloT pirdapiri da
Sebrunebuli amocana sibrtyeze, rac gacilebiT martivia.
vTqvaT mocemulia 1, 2, 3, . . . wertilebi.
1. wertilis koordinatebi x, y; direqciuli kuTxe α12 da manZili d;
saWiroa ganisazrvros x2, y2; x3, y3, . . . naxazidan x x x2 1 1= + Δ ,
x x x x x x3 2 2 1 1 2= + = + +Δ Δ Δ ,
rom davbrundeT isev 1 wertilSi
1 1 2n nx x x x x= + Δ + Δ + ⋅⋅⋅+ Δ
Δxii
n
≠=∑ 0
1
da udris Wx ,
e.i. rodesac mivediT isev 1 wertilSi Δxi
n
=∑
1
unda iyos 0, magram gvaqvs
raRac cdomileba Wx da ≠ 0 -s. marTalia kuTxeebi gavawonasworeT, magram
gverdebis sigrZeebi ar gagviwonasworebia. analogiurad y-Tvis gveqneba.
y y y2 1 1= + Δ
y y y y y y3 2 2 1 1 2= + = + +Δ Δ Δ ,
. . . . . . . . . . . . . . . . . . . .
y y y y yn n= + + + ⋅ ⋅ ⋅ +1 1 2Δ Δ Δ
Δyii
n
≠=∑ 0
1
da Wy .
116666
xazovani Secdoma W x= da y RerZebis gaswvriv SecdomaTa
geometriul jams, e.i. naxazidan
W W Wx y= +2 2 .
Seukvreloba unda veZeboT im gverdSi, romelic paraleluri iqneba
arsebuli gverdisa. e.i. gamoTvlis dros romeli gverdis r (rumbic)
miuaxlovdeba aRebuli gverdis rumbs im gverdze movaxdenT xelmeored
(gazomvas) dakvirvebas.
ganazomTa Secdomebis Teoria gvikarnaxebs, rom Tu piroba W
d∑≤
12000
(grZivi Secdoma) daculia, maSin yvelaferi sworadaa Tu W
d∑>
12000
, maSin
naxazidan ganvsazRvravT r (rumbs). naxazze patara samkuTxedidan tgrWW
y
x
= ;
amis Semdeg veZebT, romeli ganazomia saeWvo e.i. miviReT saeWvo gverdis
rumbi. igi Cvens SemTxvevaSi (ix. nax.235) aris 4-5. e.i. am xazs gavzomavT
xelmeored. axla vTqvaT, gvaqvs raRac cdomileba TiToeul metrze x
RerZis gaswvriv erTi gverdisaTvis, Secdoma TiToeuli metrisaTvis amave
RerZisaTvis unda gavamravloT gverdis sigrZeze da SevitanoT
Sebrunebuli niSniT. e.i.
Wd
ax
∑= ,
Wd
by
∑= − .
Δ Δx ad x1 1 1+ − =( ) ' Δ Δy bd y1 1 1+ + =( ) '
− ⋅ ⋅ ⋅ −ad x1 1Δ ' + ⋅ ⋅ ⋅ +bd y1 1Δ '
Δ Δx ad x2 2 2+ − =( ) ' Δ Δy bd y2 2 2+ + =( ) '
− ⋅ ⋅ ⋅ −ad x2 2Δ ' + ⋅ ⋅ ⋅ +bd y2 2Δ '
Sesworebis Semdeg gansazRvrul koordinatebs davitanT qaRaldze.
naxazidan
Δx d= cosα , Δy d= sinα ,
an λ λ λg x gd gΔ = + cosα . λ λ λg y gd gΔ = + sinα .
Sebrunebuli geodeziuri amocana ki ase gadawydeba
moc. 1 (x1, y1); 2 (x2, y2);
tgy yx x
yx
α =−−
=2 1
2 1
ΔΔ
d x y x y= = = +Δ Δ
Δ Δcos sin
( ) ( )α α
2 2
116677
nax. 160
nax. 161
13.8. wvliladebis gadatana
wvliladebis datanis mravali xerxi arsebobs: 1) marTobebis, 2)
polaruli xerxi, 3) gadakveTis (xazovan_kuTxuri), 4) gaswvrivobis xerxi, 5)
fargis sigrZivi svlis xerxi.
wvliladebis agegmva gulisxmobs, Cvens mier gamoTvlili
davukavSiroT garemos wvliladebs.
a) marTobebis xerxi.
vTqvaT, poligonis 1-2 gver-
dTan miiklakneba mdinare,
meore mxares aris saxli, Wa.
naxazze davuSvebT marTobebs
mdinaris damaxasiaTebeli
wertilebidan. davawerT x1, y1,
x2, y2, x3, y3. aq masStabi ar
gamoiyeneba. abrisi keTdeba
umasStabod, saxeldaxelod
mdinaridan xazze vuSvebT marTobebs, saxlze ki TviT xazidan ekeris
saSualebiT aRvmarTavT marTobebs im gverdze, romelic xazidan kargad
Cans. aq aseTi 2 wertilia. am
wertilebiT iqneba mimagrebuli
poligonis gverdze (1-2), manZili
saxlis kuTxeebs Soris unda
gaizomos λ λ λ λ1 2 3 8, , ⋅ ⋅ ⋅ ⋅ ⋅ .
b) polaruli xerxi. iaraRiT
vdgebiT 3 wertilSi da vmzerT
topogaremos damaxasiaTebel
wertilebs. aq I-ad izomeba 4-e
wertili, Semdeg daitkecneba
limbi, aiSveba alidada da aiReba
sami maCvenebeli. mimarTuleba-β
anaTvali lartyaze - n da daxris
116688
nax. 162
nax. 163
kuTxe ν . sawyisad miiReben 3-4-s da mimarTaven nulovan anaTvals 4-ken.
auSveben wredalidadas da damzeren Semdeg damaxasiaTebel wertils (muSa
yovelTvis dgeba gardatexis wertilebze).
g) gadakveTis xerxi. 1) kuTxuri. moc. β1 da β2 (bipolaruli
koordinatebi). A wertili ganisazRvreba β1 da β2 ; A( , )β β1 2 an direqciuli
kuTxe A( , )α α1 2 (nax. 162).
2) xazovani. vTqvaT
agegmvas vawarmoebT qalaqSi.
mocemulia kvadrati da gva-
interesebs B Wa ra manZi-
liTaa daSorebuli an sad
mdebareobs. amisaTvis vzo-
mavT A Senobidan λ λ λ1 2 3, ,
manZilebs. arsebuli sami sig-
rZiT SeiZleba gadavitanoT B
wertili gegmaze, magram kontrolis mizniT kidev gavzomavT II-re A1
wertilidanac λ λ λ' , ' , '1 2 3 da gavakontrolebT B-s gadatanis siswores.
3) gaswvrivobis xerxi (nax. igive). vTqvaT, gvinda C-Wa davitanoT
gaswvrivobis xerxiT, amisaTvis vzomavT manZils A" -sa da C-s Soris. Semdeg
A -sa da C-s gaswvrivobaSi C-dan manZils D-mde. gazomili manZilebiT
Tavisuflad davitanT gegmaze A da D-s cnobil wertilebs Soris C-
wertils.
4) fargis sigrZis svlis xerxi.
aq awarmoeben saridan saramde svlas
anu fargis gaswvriv svlas. aq fargiT
viRebT raRac cnobil adgils (vTqvaT
gzas) 9-8 da 9-10. izomeba β1 da β2
kuTxeebi da manZili 1 wertilamde d1.
Semdeg 1-Si marjvena an marcxena kuTxe
(umjobesia orive) kontrolis mizniT
aseve 2 wertilSi da ase Semdeg (nax.
164).
116699
aRniSnuli xerxi mocemuli masStabisaTvis unda iqnas SerCeuli
adgilmdebareobis mixedviT. es Cvenzea damokidebuli.
nax. 164
117700
XIV Tavi
niveloba
niveloba aris moqmedebaTa erToblioba, romlis saSualebiTac
ganisazRvreba wertilebs Soris simaRleTa sxvaoba (aRmateba).
Cven ganvixilavT mxolod sami saxis nivelobas: trigonometriuls,
geometriulsa da barometruls.
14.1. trigonometriuli niveloba
trigonometriuli niveloba sruldeba daxrili sxiviT,
Teodolitis an taqeometris saSualebiT.
nax. 165
vTqvaT saWiroa A da B wertilebs Soris simaRleTa sxvaobis
gansazRvra. warmovidginoT, rom A – wertili aris ZiriTad donebriv
zedapirze.
A-wertilSi davcentravT Teodolits.
117711
B-wertilSi ki Sveulad vayenebT lartyas. davuSvaT, rom iaraRma
mogvca ZiriTadi donebrivi zedapiris paraleluri sxivi A P' ' . davumiznoT
lartyas T' wertilSi, lartyis simaRle aris υ .
υ = BT' , xolo
instrumentis simaRle udris i-s, maSin
h i P T'= + −' υ ,
magram, radgan instrumenti ver gvaZlevs mrud xazs (ZiriTadi
donebrivi zedapiris paralelurs), aramed gvaZlevs raRac A P' -xazs da
kuTxur Secdomas ε1 -s, P P K' = , dedamiwis simrudis gamo, Secdomas. amave
dros refraqciis gamo sxivs T' -is magivrad T-Si miviRebT, T'T r= aris
refraqciis gavlena. < ε r -refraqciis kuTxuri Secdomaa, maSasadame h-is
namdvili mniSvneloba gveqneba:
h i K h r= + + − −' υ , (14.1.1)
gamovTvaloT K-s mniSvneloba. amisaTvis SevTanxmdeT, rom
R = 6400km; i = 15. m. SegviZlia instrumentis simaRle vugulvebelvyoT da
h i+ CavTvaloT R-ad.
naxazidan
( )R
LR
RL
RRRL
RRLRRCPK22
1120
2
20
2
202
02
21
21
=−⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅⋅++=−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=−+=−= ;
e.i. dedamiwis simrudis gamo Secdoma K udris iaraRidan
lartyamde manZilis kvadrats, gayofils gaorkecebul radiusze.
axla ganvsazRvroT r – refraqciis gavlena. eqsperimentalurad
dadgenilia, rom refraqciis simrudis radiusi R Rr = 6 (e.i. udris
gaeqvskecebul radiuss), maSasadame
rLR
LRr
= =02
02
2 12.
dedamiwis simrudisa da refraqciis erToblivi gavlena avRniSnoT
f -iT.
f K r= − , maSin fLR
LR
LR
= − =02
02
02
2 120 42. ;
Tu manZili L02 200= metramdea, maSin f -is gavlena mxedvelobaSi ar
aris misaRebi.
axla ganvsazRvroT h' . SevitanoT aRniSvna < =A TC' γ , maSin naxazidan
117722
γ ν ε ν ε= − + − = − +180 90 90ο ο ο( ) ( ) ,
h LL L
L
L tgtg tg
L tg
' sinsin
sinsin[ ( )]
sincos( )
sincos cos sin sin
cos ( )
= =− +
=+
=⋅ − ⋅
=
=− ⋅
≈
00 0
0
0 0
901
1
νγ
νν ε
νν ε
νν ε ν ε
νε ν ε
ν
ο
e.i. h L tg'≈ 0 ν ; gavyoT cos cosν ε⋅ radgan L0 200= m; ε -mcire kuTxea,
cosε = 1; tgε = 0 . Tu SevitanT gamoTvlil mniSvnelobebs (14.1.1) formulaSi
gveqneba
h L tg i f= + − +0 ν υ , (14.1.2)
(14.1.2) Aaris trigonometriuli nivelobis formula, romelsac aseve
geodeziuri nivelobis formulasac uwodeben. im SemTxvevaSi, rodesac L0
gazomilia bafTiT, an gamoTvlilia trigonometriuli formulebiT, gvaqvs
trigonometriuli niveloba, xolo, Tu manZilebs manZilmzomis saSualebiT
vzomavT, maSin gveqneba taqeometriuli nivelobis formula, romelsac aqvs
Semdegi saxe:
L Kn C Kn C0 12
12= + ⋅ ≈ +cos cos ( ) cosν ν ν ,
sadac L0 -aris dedamiwis fizikuri zedapiris A da B wertilebs
Soris Tarazuli manZili. K Ct= ; CFP
= 1 . K aris manZilmzomis koeficienti;
F1 -obieqtivis mTavari safokuso manZili; P – samanZilo Zafebs Soris
manZili; t-danayofis fasi; n1 -danayofis raodenoba lartyaze; ν -kuTxe,
romelsac adgens daxrili xazi mis proeqciasTan (meore ν lartyis ara
marTobulobiT Seqmnili kuTxe), xolo 1c F δ= + aris obieqtivis mTavari
safokuso manZilisa da obieqtivis linzis centridan Wogris brunvis
RerZamde manZilebis jami.
14.2. geometriuli niveloba
geometriuli niveloba ewodeba nivelobas Tarazuli sxivis
saSualebiT.
vTqvaT, gvsurs A da B wertilebs Soris simaRleTa sxvaobis gageba.
iaraRiT vdgebiT A da B wertilebs Soris, xolo A da B wertilebSi ki
117733
nax. 166
Sveulad vayenebT lartyebs. Cven, rom saSualeba gvqondes donebrivi
zedapiris paraleluri sxivis miRebis, maSin aRmateba h gamoiTvleboda
formuliT
h a b= −1 1 (14.2.1)
(sadac a1 - aris danamzeri ukana
lartyaze, b1 - aris danamzeri wina
lartyaze).
magram, iaraRi ZiriTadi doneb-
rivi zedapiris paralelur xazs ver
gvaZlevs, aramed gvaZlevs Tarazul
sxivs. e.i. dedamiwis simrudisagan
gamowveuli Secdomebi gveqneba K1 da
K2 , magram agreTve viciT, rom Tara-
zuli sxivi gaivlis ra atmosferos
sxvadasxva simkvrivis fenebSi, refraq-
ciis gamo gadatydeba da sabolood
sxivi miiRebs mimarTulebas, romelic
naxazze mTliani xaziT aris aRniS-
nuli. refraqciis gavlena iqneba r1
da r2 . axla ganvsazRvroT a1 da b1 .
a a k r a f1 1 1 1= − − = −( ) avRniSnoT ( )k r f1 1 1− − -iT,
b b k r b f1 2 2 2= − − = −( ) avRniSnoT ( )k r f2 2 2− − -iT.
SevitanoT (1) gamosaxulebaSi a1 da b1 -is mniSvnelobebi. gveqneba
h a f b f a b f f= − − − = − + −( ) ( ) ( )1 2 2 1 (14.2.2)
dadgenilia, rom fdR1
10 42= . ; fdR2
20 42= . ,
sadac d1 -aris A wertilidan iaraRis centramde manZili; d2 -aris B
wertilidan iaraRis centramde manZili; R-aris dedamiwis radiusi. im
SemTxvevaSi, roca d d1 2= , maSin f f1 2= da r r1 2≈ (daaxloebiT tolia imitom,
rom manZilebi sxavadasxvaa).
h a b= − , (14.2.3)
maSasadame, rodesac iaraRiT manZilis SuaSi vdgebiT, refraqciisa da
simrudis gavlena ispoba. im SemTxvevaSi, roca d1 0= maSin
117744
nax. 167
h i b f= − −( ) , (14.2.4)
(i – aris iaraRis simaRle). rogorc vxedavT, ufro mosaxerxebelia niveloba
Suidan (garda amisa Suidan warmoebuli nivelobis dros ispoba
instrumentaluri Secdomebi).
vTqvaT vawarmoebT A da B wertilebs Soris nivelobas. vdgavarT am
wertilebs Soris ara tol manZilze, e.i. ara zustad SuaSi. iaraRi
nacvlad Tarazuli sxivisa gvaZlevs daxril sxivs, amitom simaRleTa
sxvaoba toli iqneba
h a a b b= + − +( ) ( )Δ Δ , (14.2.5)
(sadac a da b anaTvlebia
lartyaze), roca d d1 2= , maSin
Δ Δa b= da h a b= − e.i. ispoba
instrumentaluri Secdomebis
gavlena.
rTuli niveloba (geometriuli)
rTuli niveloba aris martivi nivelobebis jami. misi arsebobis
mizezia: 1) wertilebs Soris didi manZili, 2) dabrkoleba wertilebs Soris,
3) simaRleTa didi sxvaoba wertilebs Soris.
vTqvaT, saWiroa vawarmooT A1 da A n wertilebs Soris simaRleTa
sxvaobebis gansazRvra.
nax. 168
H h h h a a a b b b a bn n n i ii
n
i
n
= + + ⋅⋅ ⋅ + = + + ⋅ ⋅ ⋅ + − + + ⋅ ⋅ ⋅ + = −==∑∑( ) ( ) ( )1 2 1 2 1 2
11
14.3. nivelirebi
h a bh a b
h a bn n n
1 1 1
2 2 2
= −= −
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅= −
;;
.
117755
nax. 169
nivelirebi ZiriTadad ori tipis gvxvdeba: 1) xSuli da 2) gadasadeb
Wogriani.
1) xSul nivelirebSi TviT Wogrzea mimagrebuli kontaqturi
Tarazoebi da misi boloebi, rodesac Tarazos RerZi ar aris Tarazulad
moyvanili, gamoCndeba ase , Tarazul mdgomareobaSi ki . 2)
gadasadeb Wogrian nivelirebSi Tarazoebi gvxvdeba a) SimSaze mimagrebuli;
b) Wogrze mimagrebuli da Wogrze dasadgmeli.
a) Semowmebebi
orive tipis nivelirebs aqvT sami saerTo
Semowmeba.
I-li Semowmeba. ZafTa badeze Sveuli Zafi unda iyos mkacrad Sveulad,
rom Tarazuli Zafi iyos Tarazulad. Tarazo moyavT SuaSi. niveliridan
30-50m-is daSorebiT kedelze daniSnaven wertils, romelic Sua ZafiT
ifareba (aiReben anaTvals lartyaze), Semdeg mikrometruli xraxniT nela
amoZraveben nivelirs vertikaluri RerZis garSemo. Sua Zafi unda gayves am
wertils (an ar icvlebodes anaTvali lartyaze), Tu es ar sruldeba, maSin
Seasworeben ZafTa badis Semasworebeli xraxniT. es Sesworeba xraxnebis
saSualebiT sruldeba, magram rTuli Sesworebaa da axali sistemis
iaraRebSi amas qarxana uzrunvelyofs.
II – Semowmeba. Wogris optikuri Zala da Tarazos mgrZnobiaroba
urTierT Sesabamisi unda iyos. viciT, rom Wogris optikuri Zala TG
=60"
(G -gamadideblobaa, 60" -mzeris kritikuli kuTxe). cnobilia, rom buStulis
nul punqtze dayenebis zRvruli sizuste= 015. τ (sadac τ -Tarazos danayofis
fasia), e.i. 60 015" .G
= τ ; Gτ = 400 . gamartivebuli saxe am Semowmebisa
mdgomareobs SemdegSi: ayeneben lartyas niveliridan 50-70m-is daSorebiT.
moyavT Tarazo SuagulSi da iReben anaTvals lartyaze. Semdeg xraxniT
buStulas gadaadgileben Suidan da Semdeg isev imave xraxniT moyavT
SuaSi. iReben anaTvals. Tu anaTvali igive iqneba, maSin Tarazos
mgrZnobiaroba Seesabameba Wogris gamadideblobas an metia masze. Tarazos
zedmet sizustes amJRavneben Semdegnairad: Tarazo moyavT SuaSi, iReben
117766
nax. 170
anaTvals a1-s, Semdeg damyenebeli an Semasworebeli xraxniT cvlian
anaTvals da kvlav abruneben a1 anaTvalze. Tu buStula kvlav SuaSi ar
dadga, maSasadame, igi zedmeti sizustis yofila. zedmeti sizustis Tarazo
ki sagrZnoblad anelebs muSaobis temps.
b) Tarazos safasuris gansazRvra lartyis saSualebiT
vake adgilze niveliridan ≈ 50 m-is daSorebiT kargad ganaTebul
adgilze ayeneben lartyas (niveliris erTi Semasworebeli xraxni lartyis
mimarTulebiT unda iyos), d manZils niveliridan lartyamde zomaven
xveulaTi an bafTiT.
Wogrs umizneben lartyas,
amwevi xraxniT Tarazos buStu-
las gadaweven Suidan, mxolod
ise, rom orive bolos anaTvlis
aReba SeiZlebodes. iReben anaT-
vals lartyaze b1 da buStulis
boloebze a1 da a2, Semdeg
buStulas gadaadgileben meore mxares da kvlav iReben anaTvlebs
lartyaze b2 da buStulis boloebze a3 da a4 gamoiTvlian λ = −b b2 1 .
buStulis boloebiT aRebuli anaTvlebis sizustis kontrolisaTvis
gamoTvlian buStulis sigrZes a a2 1− da a a4 3− , maT Soris dasaSvebia
gansxvaveba 0.2 – 0.3 danayofi. Semdeg gamoiTvleba buStulis mdebareoba
orive gadaxrisaTvis na a
11 2
2=
+ da n
a a2
3 4
2=
+ da buStulis gadaxras n-s,
n n n= −2 1 .
naxazidan naTlad Cans, rom Wogris mzeris RerZis Setrialebas ε
mcire kuTxeze, Seesabameba Tarazos buStulis n danayofze gadaadgileba.
amitom erTi danayofis safasuri τε
=n, ε mcire kuTxe SeiZleba CaiTvalos
centralur kuTxed, romelic d radiusis mqone λ λ( )= −b b2 1 rkals Wimavs,
amitom ε ρ=λd
" da τ -ki gveqneba τ ρ=⋅λ
d n" . xSirad Tarazos safasurs
sazRvraven xelsawyoTi – egzamenatoriT.
117777
nax. 171
III – Semowmeba. Wogris fokusirebis dros mzeris RerZi ar unda
tydebodes. vake adgilze SeirCeven C wertils da am wertilidan ≈ 50 m-is
radiusiT daniSnaven A1, A2, A3,
A4, A5 wertilebs. C wertilSi
ayeneben nivelirs da iReben
anaTvlebs lartyaze A1, A2, A3,
A4, A5 wertilebSi. vinaidan es
wertilebi C wertilidan toli
manZilebiT arian daSorebuli (r=50m), amitom Wogris fokusireba mxolod
erTxel A1 wertilze gvWirdeba. gamoviTvliT simaRleTa sxvaobebs
(aRmatebebs), am wertilebs Soris h1, h2, . . . h4-s, Semdeg gadagvaqvs iaraRi C1
wertilSi. C1 mdebareobs AC mimarTulebis gagrZelebaze≈ 10 m-is daSorebiT.
C1 wertilidan daniSnul (A1, A2, A3, A4, A5) wertilebze anaTvlebis asaRebad
saWiro xdeba yvela danamzerze fokusireba, vinaidan manZilebi C1
wertilidan daniSnul wertilebamde sxvadasxvaa. viRebT anaTvlebs da
vsazRvravT simaRleTa sxvaobebs h h h' , ' ,... '1 2 4 . cxadia simaRleTa sxvaobebi
gansazRvruli C-wertilidan, toli unda iyos simaRleTa sxvaobebisa C1-
wertilidan (an gansxvavdebodes 2τ -Ti, sadac τ -lartyaze anaTvlis aRebis
sizustea). e.i. h h1 1 2− ≤' τ ; h h2 2 2− ≤' τ ; h h3 3 2− ≤' τ ; h h4 4 2− ≤' τ ; Tu es piroba
daculia, maSin cxadia, rom niveliris mzeris RerZi ar tydeba fokusirebis
dros. Tu es piroba ar aris daculi, maSin aseTi niveliriT sargebloba
mxolod im SemTxvevaSia SesaZlebeli, Tu SevasrulebT nivelobas SuaSi
dgomiT. gadasadeb Wogrian nivelirebisaTvis es Semowmeba sruldeba
ormagi nivelobiT. es igivea rac xSuli nivelirebis meore Semowmeba (ix.
qvemoT).
g) xSuli nivelirebis Semowmeba
I – Semowmeba. Tarazos RerZi marTobi unda iyos iaraRis brunvis
RerZisa.
Tarazos aTavseben ori damyenebeli xraxnis gaswvriv da orive
xraxnis urTierT sawinaaRmdego brunviT buStula moyavT SuaSi (H1H2),
Semdeg Wogrs Semoabruneben 180ο-iT (TvaliT) (H H' '2 1 ). Tu buStula
gadaixara, maSin gadaxris naxevars Seasworeben Tarazos Semasworebeli
117788
xraxniT, Semdeg buStula SuaSi mohyavT damyenebeli xraxnebiT. zemoT
aRweril operacias imeoreben manam, sanam yoveli 180ο-iT Semobrunebis dros
buStula ar gaCerdeba SuaSi (gadaxra dasaSvebia erTi danayofis 0.2-0.3)
(H H" "2 1 ). Semdeg Semoatrialeben Wogrs 90ο -iT da mesame damyenebeli
xraxniT Tarazos buStula moyavT SuaSi. axla sibrtyeze mdebare ori xazi
Tarazuladaa, e.i. sibrtye Tarazuladaa. amis Semdeg brunvis RerZi sakmaod
kargad aris Sveulad moyvanili. magram Tarazos Semowmebas axdenen
xelaxla da sabolood brunvis RerZi moyavT Sveul mdgomareobaSi.
II – Semowmeba. Wogris mzeris RerZi paraleluri unda iyos Tarazos
RerZisa. aRniSnuli Semowmeba sruldeba ormagi nivelobiT, amasTan
SeiZleba gamoviyenoT erT-erTi meTodi arsebuli ori meTodidan.
I-li meTodi. nivelirs dgamen jer A wertilSi ise, rom okulari da A
wertilSi Camagrebuli palo erT vertikalur sibrtyeSi iyos. moyavT
iaraRi momwesobaSi, sazRvraven iaraRis simaRles palodan okularis
centramde i1 -s. B wertilSi paloze adgamen lartyas da iReben anaTvals a1-
s. Semdeg gadaaqvT iaraRi B wertilSi da lartya A wertilSi.
analogiurad ganisazRvreba iaraRis simaRle i2 da anaTvali lartyaze a2.
β ε ε ε= + − − =( ) ( )90 90 2o o ε β
=2
nax. 172 nax. 173
117799
nax. 176
a i hi= −
nax. 174 nax. 175
h i a a= − +1 1( )Δ h a a i= + −( )2 2Δ
gamoTvlian 1 2 1 2
2 2i i a aa + +
Δ = − ,
Tu Δa ≤ ±2 mm, maSin piroba daculia, Tu gansxvavebulia, maSin unda
Seswordes ZafTa bade.
II meTodi. moedanze paloebiT daniSnaven tolgverda samkuTxeds
≈ −40 50 metriani gverdebiT. Suidan nivelobiT C wertilSi dgomiT
ganisazRvreba A da B wertilebs Soris simaRleTa sxvaoba h. moqmedeba
unda Sesruldes zustad, ara nakleb 2-jer, ganazomis SecdomiT ara umetes
±2 mm-isa. amis Semdeg niveliri gadaaqvT A wertilSi (an B-Si). sazRvraven
niveliris simaRles palodan okularis centramde. winaT Sesrulebuli
nivelobis SedegiT cnobilia aRmateba h. advili gamosaTvlelia, rom
anaTvali lartyaze unda iyos a i h= − , amis Semdeg iReben anaTvals
lartyaze a1-s B wertilSi. Tu sxvaoba a a1 2− ≤ mm, maSin piroba daculi
iqneba, Tu ara da maSin Seswordeba ZafTa bade vertikaluri Semasworebeli
xraxnebiT.
118800
nax. 177
nax. 178
d) gadasadeb Wogriani nivelirebis Semowmeba
I – Semowmeba: Wogris mzeris RerZi unda emTxveodes geometriul (da
optikur) RerZs.
aRniSnuli Semowmebis Sesas-
ruleblad dgamen nivelirs, moyavT
misi RerZi Sveul mdgomareobaSi
da iaraRidan ≈ −50 70m-is daSo-
rebiT aTavseben lartyas da
iReben anaTvals a1-s. Semdeg gada-
atrialeben Wogrs Tavis budeSi
180ο-iT da kvlav iReben anaTvals
a2-s. Tu anaTvali igivea, maSin piroba Sesrulebulia, Tu gadaadgilda, maSin
piroba darRveulia da asworeben Semdegnairad: gamoTvlian aa a
=+1 2
2
mniSvnelobas da ZafTa badis Semasworebeli xraxnebiT Seasworeben. ZafTa
badis horizontaluri Zafi ar gviCvenebs a-s tol anaTvals.
II – Semowmeba: aucilebelia, rom Wogris salteebis msaxveli (ab)
paraleluri iyos Tarazos RerZis (H1H2), sasurvelia aseTive
damokidebuleba iyos Tarazos RerZisa (H1H2) da Wogris mzeris RerZs
Soris.
aq SeiZleba Segvxv-
des sami SemTxveva
1) d1=d2 da P1=P2
2) d1=d2 da P1 ≠ P2
3) d1 ≠ d2 da P1 ≠ P2
pirvel SemTxvevaSi
Tarazos RerZi Wogris
salteebSi gadadebis Sem-
deg horizontalurad darCeba. aq H H1 2ÈÈab da H H1 2ÈÈKT, e.i. moTxovnili
piroba Sesrulebulia. meore SemTxvevaSi, Wogris gadadebis Semdeg Tarazos
buStula gadaixreba, radganac Tarazos RerZsa da salteebis msaxvels
Soris arsebobs kuTxe. ra Tqma unda, Tarazos SuaSi mosayvanad saWiroa
buStula gadavaadgiloT vertikaluri Semasworebeli xraxnebiT, gadaxris
118811
naxevriT. Semowmeba sruldeba ramdenjerme. ris Sedegadac vaRwevT, rom
P1=P2 da amasTan H H1 2ÈÈabÈÈKT pirobasac. mesame SemTxvevaSi, roca
diametrebi ar aris toli, aRniSnuli SemowmebiT vaRwevT H H1 2ÈÈab, magram
Tarazos RerZi H H1 2 da geometriuli RerZi KT paralelurebi ar iqnebian.
garda amisa, am Semowmebis ganxilvis dros Cven davuSviT, rom Tarazos
RerZi da geometriuli RerZi erT sibrtyeSi mdebareoben, magram, SeiZleba,
rom es ase ar iyos. Semowmebis Sesasruleblad Tarazos buStula moyavT
nul punqtSi, Semdeg arxeven Tarazos, nel-nela atrialeben Wogrs
salteebSi. Tu buStula sagrZnoblad gadaixara Suidan, maSin mas
asworeben Tarazos gverdiTi Semasworebeli xraxnebiT. Semdeg aRniSnul
Semowmebas imeoreben. amasTan erTad, Tarazos rxevis Semdeg Wogris brunvis
dros salteebSi SeiZleba SevniSnoT buStulis sami mdebareoba. 1)
buStula adgilze (nul punqtze) rCeba. 2) Tarazos buStula nulpunqtidan
sul erT mxares ixreba. 3) buStula xan erT mxaresaa, xan meore.
pirvel SemTxvevaSi Tarazos RerZi paraleluria geometriuli RerZis,
meore SemTxvevaSi Tarazos RerZi da geometriuli RerZi erT sibrtyeSi
mdebareoben, magram paralelurebi ar arian. mesame SemTxvevaSi es piroba
daculi ar aris, mis misaRwevad Tarazos gverdiTi Semasworebeli xraxnebiT
unda iqnas miRweuli is, rom buStula ar gadadiodes nul punqtis erTi
mxridan meore mxareze.
14.4. nivelobis warmoeba
daniSnulebis mixedviT niveloba aris: gaswvrivi, gaswvriv-ganivi da
moednis.
a ) g a s w v r i v - g a n i v i n i v e l o b a sruldeba zoluri reliefis
asaxva-Semecnebis mizniT, sanam geometriul nivelobas SevasrulebdeT am
zolze, winaswari kvleviTi samuSaoebis safuZvelze (barometruli an
taqeometriuli samuSaoebis gamoTvlebis Semdeg), Seswavlilia sanivelo
zoli.
am mizniT warmoebuli niveloba ori ZiriTadi amocanisagan Sedgeba: 1)
sanivelo RerZis Semzadeba da 2) TviT nivelobis warmoeba.
118822
nax. 179
sanivelo RerZis Sesamzadeblad, saWiroa gvqondes kuTxzomiTi iaraRi
(Teodoliti), mrudeebis dakvalvis cxrili, sapiketaJo Jurnali, 20-metriani
foladis bafTa, 6 an 11 foladis Cxiri, xveula, 3 – 4 sari, paloebi, fanqari,
najaxi. sapiketaJo Jurnali warmoadgens
milimetrebian wignaks, romelic ivseba
Semdegnairad. upirvelesad irCeven tra-
sis mimarTulebas da amagreben niSnebiT
trasis sawyis da bolo wertilebs,
moxvevis wertilebs da a.S. Semdeg
awarmoeben trasis piketaJs, e.i. piketebis
daniSvnas. piketi aris palo, romliTac
trasis wertilebi aRiniSneba. piketebs
amagreben erTi meoridan 100 metris
daSorebiT (qalaqisa da sawarmoo
teritoriaze misi daSoreba SeiZleba
iyos 40-50 metri). manZilebs piketebs Soris zomaven foladis bafTisa da
Cxirebis daxmarebiT. garda amisa piketebs Soris manZili damokidebulia
imaze, Tu ramdenadaa saWiro trasis reliefis Seswavla. piketebis
numeracia iwyeba trasis sawyisi wertilidan. TviTeuli sapiketo wertili
aRiniSneba ori paloTi: “wertiliT da darajiT”, wertilSi miwis piramde
Caismeba palo 10 sm-is siRrmeze da nivelobis warmoebis dros masze
idgmeba lartya. wertilidan 10 sm-is daSorebiT Caisoba meore palo
miwidan 10 sm-iT zeviT amosuli. es aris e.w. “daraji”, romelic gamoiyeneba
wertilis mosaZebnad. masze fanqriT iwereba piketis nomeri. Sualedi
wertilebi, romlebic saWiroa damatebiT reliefis dasaxasiaTeblad,
aRiniSneba mxolod “darajiT” da waewereba wina piketis nomers plius
daSoreba am piketidan Sualed wertilamde. mag: (2)+40, amitom aseT
wertilebs plius wertilebi ewodeba. ganiv wertilebsac aRniSnaven
mxolod “darajebiT” da waewereba maTze nomeri da daSoreba trasidan.
b ) m o e d n i s ( z e d a p i r i s ) n i v e l o b a .. samSeneblo moednebze,
aerodromebze, stadionebze da a.S. xSirad warmoiqmneba miwis samuSaoebis
moculobis gamoTvlis saWiroeba anda vertikaluri dagegmarebis
(nivelobis) Sesruleba, risTvisac aucilebelia msxvilmasStabiani (1 : 1000,
118833
1 : 500, 1 : 200) topo-gegmebis Sedgena kvadratebis meTodiT, Seqmnil
sasimaRlo safuZvelze dayrdnobiT.
adgilze vagebT dasakvalav kvadratebs, Tu kveTis simaRle 0,25h = m
kvadratis gverdi 10 40a = − m, roca 0,5h = m, 50 100a = − m da ufro mets.
kvadratebis numeracia mocemulia 256 naxazze:
nax. 180
adgilze qselis dakvalvis Semdeg vatarebT nivelobas. Jurnalis
nacvlad yvela monacemi gamoisaxeba qaRaldze, romelzec iwereba yvela
saWiro informacia. winamdebare naxazze (180) yoveli kvadratis centrSi
arsebuli cifrebiT naCvenebia instrumentis dgomis wertilebi, romlebic
rac SeiZleba axlos unda iyos kvadratebis diagonalebis gadakveTis
wertilebTan. moiTxoveba, rom yovel wertilze iyos minimum ori danamzeri,
amitom (180) naxazze 19 sadguria da amiT dakmayofilebulia moTxovnili
piroba. kvadratebis niveloba damokidebulia misi gverdebis sigrZeze. Tu
100a ≥ m, maSin yoveli kvadratis niveloba tardeba dgomis erTi
wertilidan; Tu 100a < m, maSin erTi sadguridan nivelobas vawarmoebT
ramodenime kvadratze. (180) naxazze vixilavT SemTxvevas, roca 100a ≥ .
niveliris gadaadgilebis sqema naCvenebia punqtiriT.
nivelobis Catareba kidev SeiZleba paraleluri xazebiTac. saWiroa
Seiqmnas gaxsnili an Sekruli svla, raTa vuzrunvelvyoT dakvirvebaTa
Sedegebis kontroli.
a
11 22 33 44 55 66 AA
BB
AA
BB
CC
DD
EE
FF
XX VV XXVVIIII XXVVIIIIII XXIIXX VVIIII
II IIII IIIIII IIVV VV
XX II II II XXIIII XXII XX IIXX
XX VV II VVII
XX II VV VVIIIIII DD44
DD
EE
FF
a
118844
kameraluri damuSaveba gamoixateba SemdegSi:
1) gamoviTvliT momijnave kvadratebis horizontebis simaRleTa sxvaobas
(sadguris): 1 1i i i iH b b H H+ += − = −V h h ,
sadac i aris kvadratis (sadguris) nomeri;
1ib + - anaTvali lartyaze, romelic dgas momdevno kvadratis wveroze;
ib - anaTvali winamdebare sadgurze;
2) vasrulebT sanivelo svlis kontrols (Tu svla Sekrulia, maSin
hH f=∑V , unda davicvaT Semdegi piroba: 2hf n= ⋅sm , sadac n aris
sadgurebis raodenoba svlaSi; 2sm=2m, rodesac anaTvali lartyaze
1sm-dea);
3) vawarmovebT sanivelo svlis gawonasworebas; cdomilebas vanawilebT
Sebrunebuli niSniT TiToeul HV -ze Tanabrad, gamoviTvliT Hh ;
4) gamoviTvliT kvadratis wveroebis niSnulebs Semdegi formuliT:
DH H d= −h ,
sadac d aris anaTvali lartyaze D wertilSi.
5) samuSao Tavdeba gegmis SedgeniTa da misi gamoxazviT.
saxazav formatze mocemuli masStabiT vadgenT kvadratebis sqemas.
miRebuli kvadratebis wveroebis niSnulebis garda, formatze unda
davitanoT yvela saSualedo wertilebi (reliefis maxasiaTebeli
wertilebi da konturebi). Jurnalidan unda amoviweroT yvela wertilis
niSnulebi (1sm-mde). Semdeg vaxdenT horizontalebis (izohifsebis)
interpolirebas da reliefis gamoxazvas (qanobebis da reliefis yvela
damakavSirebeli xazis CaTvliT). gegma unda daixazos tuSiT pirobiTi
aRniSvnebis sruli SesabamisobiT.
118855
nax. 181
14.5. mrudeebis dakvalva
nagebobaTa RerZebi ganlagebulia ara marto swori an texili
xazebis mimarTulebiT, aramed zogjer mrudi xazis mimarTulebiTac (mag.
sanaosno arxebi, gzebi, didi kaSxlebi, saxlebi). aseT SemTxvevaSi ori
sworxazovani mimarTulebis SeuRleba xdeba metwilad mrudis saSualebiT
(nax. 181).
mrudis dakvalvisaTvis saWiroa moxvevis
kuTxe α da mrudis radiusi R , rac
saSualebas gvaZlevs vipovoT adgilze
mrudis sami mTavari wertilis mdebareoba:
A1 -mrudis dasawyisi, P1 -mrudis Sua wer-
tili da D1 -mrudis bolo wertili.
mTavari wertilebis mdebareobis adgilze
gadasatanad α -moxvevis kuTxisa da R
radiusis sidideebis codnis garda
saWiroa vicodeT: A P PD1 1= mxebi xazis
sigrZe, romelsac SemoklebiT ewodeba T (tangesi), A D1 1 -mrudis sigrZe,
romelic aRiniSneba K-Ti. PP1 -s aRniSnaven Б -Ti da ewodeba biseqtrisa
(manZili kuTxis wverodan mrudis Sua wertilamde). unda vicodeT agreTve
D minazomi, romelic warmoadgens ori mxebis da mrudis sigrZis sxvaobas.
e.i. D T K= −2 . simrudis radiusi winaswaraa mocemuli, xolo α -s
ganvsazRvravT TeodolitiT. gamovTvaloT mrudis damaxasiaTebeli
elementebi. naxazidan
T R tg= ⋅α2 (14.5.1);
RTR
SecRROPБ2
2cos
4sin2
12
22
≈=⎟⎠⎞
⎜⎝⎛ −=−=
α
αα
(14.5.2);
mrudis sigrZe aris K ; 2 360
KR
απ
= o ; K R R R= = =
2360 180π α π α α
ρο ο (14.5.3);
D T K= −2 ; γ α= +90
2ο ;
piketaJis gamoTvlis Semowmeba
118866
nax. 182
vTqvaT gazomvebiT miviReT T K D= = =50 95 5m m m
(28) 10 (0) 50
(27) 60 (0) 95
(28) 55
ByT
HKK
KK
= +− = +
= ++ = +
= +
ByT
DKK
= ++ = +
= +− = +
= +
( )( )( )( )
( )
28 100 5028 600 05
28 55
Tu reliefi rTulia, maSin saWiroa ufro dawvrilebiT dakvalvis
Catareba. mrudis detalurad dakvalvisaTvis arsebobs sami xerxi:
1) marTobebis xerxi
2) kuTxeebis xerxi
3) wagrZelebuli qordebis xerxi
a) marTobebis xerxi
vTqvaT mrudze, romlis radiusia R, saWiroa P P P1 2 3, , ... wertilebis
daniSvna ise, rom maT Soris mrudwiruli manZili udrides K-s (nax. 182). am
mizniT AM mxebi miviRoT abscisis RerZad, AO ordinatis RerZad. A
wertili ki RerZis dasawyisad. maSin, P P P1 2 3, , ... wertilebis mdebareoba
mrudze SeiZleba ganisazRvros marTkuTxa koordinatebiTY( x y1 1, ), ( x y2 2, ),
( x y3 3, ), am mizniT ganvsazRvroT K-mrudwiruli manZilis Sesabamisi ϕ
kuTxis sidide. gasagebia, rom
ϕπK R
=3602
ο
aqedan ϕπ
= ⋅KR
180ο
(14.5.4)
(nax. 182)-dan gveqneba
mcire radiusis Sem-
TxvevaSi X-abscisebis mniS-
vnelobebs gadavzomavT
mxebis mimarTulebiT,
xolo B, C da D werti-
lebidan aRmarTuli mar-
X R1 = sinϕ ; 21 cos (1 cos ) 2 sin
2y R R R R ϕϕ ϕ= − = − = (14.5.5)
X R2 2= sin ϕ ; y R R R R222 1 2 2= − = − =cos ( cos ) sinϕ ϕ ϕ (14.5.6)
118877
nax. 183
Tobebis mimarTulebiT y1, y2 da y3 koordinatebis gamoTvlil
mniSvnelobebs. mrudze miRebuli wertilebi iqneba: P1, P2, P3.
roca radiusi R didia, maSin X abscisi Sesabamisi rkalisgan mcired
gansxvavdeba. amitom, ufro mosaxerxebelia praqtikulad X RerZis gaswvriv
gadavzomoT ara abscisTa mniSvneloba, aramed e.w. „mrudi uabscisod“. K-X.
amisaTvis X RerZze gadaizomeba K-rkalis sigrZe da Semdeg sawinaaRmdego
mimarTulebiT (ukan) movzomavT rkalis bolo wertilidan K-X sigrZes da
miRebul wertilSi abscisaTa RerZis marTob mimarTulebaze xveulaTi
gadazomaven y sigrZes. miRebuli wertili iqneba mrudze.
marTobebis xerxiT mrudze P1, P2, P3... wertilebi miiReba erTmaneTisagan
damoukideblad. amis gamo Secdomebis dagroveba ar xdeba. am xerxis
Rirsebac swored amaSi gamoixateba.
b) kuTxeebis xerxi
es meTodi gamoiyeneba im
SemTxvevaSi, roca dakvalva mar-
TkuTxa koordinatebis wesiT mou-
xerxebelia. am meTodis arsi Sem-
degSi mdgomareobs: vTqvaT, gvinda
mrudis dasawyisidan, romlis radi-
usia R, aRvniSnoT maszed ramodenime
wertili A, B, C, D, romlebic
daSorebuli arian erTmaneTisagan S
sigrZis qordiT (nax. 183). amisaTvis
gansazRvraven S qordis Sesabamis
centralur kuTxe ϕ -s Semdegnairad:
S R2 2
= ⋅ sin ϕ aqedan sin ϕ
2 2=
SR (14.5.7)
aqedan gamomdinare centraluri kuTxis naxevars an kuTxes, mxebsa da
mkveTs Soris gansazRvraven (7) gamosaxulebiT. Tu cnobilia S (qorda) da
R (radiusi), mrudis sawyis A wertilSi dgamen Teodolits. instruments
moiyvanen momwesobaSi. limbis nuls SeuTavseben wred-alidadis nuls da
118888
nax. 184
wred-alidadas daamagreben. auSveben limbs da Wogrs umizneben M
wertils. am dros daamagreben limbs, auSveben alidadas da AM
mimarTulebidan gadazomaven kuTxes, romelic tolia ϕ2-is da WogrSi
mzeris dros am mimarTulebiT gadazomaven qordas, riTac miiReben mrudis
B wertils. axla, Tu anaTvals limbze gaadideben ϕ2-kuTxiT, miiReben
Wogris axal mdebareobas, romelic mimarTuli iqneba C wertilisaken. ris
Sedegadac B wertilidan mozomaven Wogris mzeris RerZis mimarTulebis
gadakveTamde S (qordis) sigrZes da miiReben mrudis C wetrils. ase
agrZeleben dakvalvas saboloo Sedegis miRebamde. A, B, C, D wertilebSi
Caasoben paloebs.
am xerxis uaryofiTi mxare is aris, rom Sualedi wertilebis
raodenobis zrdasTan erTad, izrdeba mrudze wertilebis mdebareobis
gansazRvris Secdoma.
g) wagrZelebuli qordebis meTodi
am meTodiT mrudis dakvalvis
SemTxvevaSi iqcevian Semdegnairad:
mrudis dasawyis A wertilSi (nax. 184)
mxebis mimarTulebas daniSnaven Teo-
dolitiT an goniometriT. Semdeg
SeirCeven qordis sigrZes (10m an 20m)
da gamoTvlian mrudis B wertilis (x
da y) koordinatebs Semdegi formu-
lebiT
x R= sinϕ (14.5.8)
y R= 22
2sin ϕ (14.5.9)
sadac ϕ -aris S-qordis Sesabamisi centraluri kuTxe. Tu R mocemulia,
ϕ -advilad ganisazRvreba Semdegi formuliT: sin ϕ2 2
=SR, amis Semdeg
118899
saWiroa iangariSon e.w. K Sualedi gadaadgileba. misi sidide
ganisazRvreba tolferda C BC' da AOB samkuTxedebidan. am samkuTxedebSi
∠ = ∠ =C BC AOB' ϕ , e.i. isini msgavsia.
aqedan gamomdinare KS
SR
= , maSin K SR
=2
, B wertilis y koordinatis
sidide (romelsac kidur gadaadgilebebs uwodeben), Seadgens Sualedi
gadaadgilebis naxevars. e.i. y K=
2, TviT dakvalvis Tanmimdevroba adgilze
sruldeba Semdegnairad: mrudis A sawyisi wertilidan mxebis
mimarTulebaze movzomavT X-sidides da mis marTobulad ki gadavzomavT y-s.
miviRebT mrudis B wertils. Tu B wertilidan qordis mimarTulebas
gavagrZelebT da masze movzomavT S-qordis sigrZes, miviRebT C' -wertils.
axla saWiroa B-wertilidan (romelsac centrad viRebT) movxazoT S
qordis toli radiusiT rkali da C' -wertilidan xveulaTi gadavzomoT K-s
mniSvneloba. maTi urTierT gadakveTiT miviRebT mrudis saZiebel C
wertils. ase vagrZelebT dakvalvas danarCeni wertilebis mrudze misaRebad.
es xerxi sxva xerxebTan SedarebiT nakleb sizustes iZleva, magram
miuxedavad amisa, misi moxerxebulad gamoyeneba SeiZleba SezRudul
adgilebSi (mag. gvirabSi an mTagorian viwro adgilebSi).
d) serpantini
nax. 185
vTqvaT AC midamoSi SeuZlebelia dakvalvis warmoeba β kuTxis
simciris gamo da iZulebuli varT winaaRmdegobas SemovuaroT (mag. xevis
119900
an dabrkolebis gamo). amas qvia serpantini, e.i. serpantini aris sworebisa
da Sebrunebuli mrudeebis erToblioba.
velze vzomavT β -kuTxes, xolo winaswar mocemuli iqneba r, d da R.
r – ZiriTadi mrudis radiusia,
R – Sebrunebuli mrudis radiusia,
d – ZiriTad da Sebrunebul mrudebs Soris moTavsebuli swori ubani.
am radiusebs gvaZleven reliefisa da samuSaos (gza, arxi) mixedviT,
α -aris moxvevis kuTxe. rogorc ukve viciT
T Rtg=α2, K R= ⋅
αρ
, (14.5.10)
12
Б R Sec α⎛ ⎞= −⎜ ⎟⎝ ⎠
, D T K= −2 , (14.5.11)
ΔNBE -dan rtg
d Tα =
+, (14.5.12)
(14.5.10)-dan T -s mniSvneloba SevitanoT (14.5.12)-Si, miviRebT
tg rd Rtg
αα
=+ 2
. (14.5.13)
(14.5.13) gantolebis orive mxare gavamravloT ⎟⎠⎞
⎜⎝⎛ +
2αRtgd -ze miviRebT:
rRtgdtg =⎟⎠⎞
⎜⎝⎛ +
2αα rtgRtgdtg =+
2ααα , (14.5.14)
21
22
2 α
α
αtg
tgtg
−= . tgα -is es mniSvneloba SevitanoT (14.5.14) gamosaxulebaSi,
miviRebT 2
2
12
22
12
2
2
2
dtg
tg
Rtg
tgr
α
α
α
α−
+−
= .
am gantolebis orive mxare gavamravloT ⎟⎠⎞
⎜⎝⎛ −
21 2 αtg -ze, miviRebT:
⎟⎠⎞
⎜⎝⎛ −=+
21
22
22 22 ααα tgrRtgdtg ,
222
22 22 ααα rtgrRtgdtg −=+ ,
119911
2 22 2 02 2 2
dtg Rtg rtg rα α α+ + − = ,
02
22
)2( 2 =−++ rdtgtgrR αα.
amovxsnaT tg α2 -is mimarT
tgd d R r r
R rα2
22
2
=− ± + +
+( )
,
aqedan gavigebT α -s. BE r=
sinα;
β -kuTxe izomeba adgilze TeodolitiT. γ α β α β= − − − = + −360 2 90 180 2ο ο ο( )
TviT dakvalva sruldeba Semdegnairad: upirveles yovlisa, unda
ganisazRvros adgilze S mrudis mdebareoba. ZiriTadi mrudis N wertilis
misaRebad TeodolitiT vdgebiT B wveroze, Wogrs mivmarTavT A
wertilisaken da limbs davamagrebT. Semdeg marto alidadis brunviT
Wogrs SevatrialebT ( )90ο − α -Ti (vernierze saTanado anaTvlis aRebiT). am
SemTxvevaSi Wogris mzeris RerZi iqneba mimarTuli N wertilisaken da Tu
am mimarTulebiT gadavzomavT r-radiusis sigrZes, miviRebT TviT N wertils.
analogiurad, BC xazTan dakavSirebiT miviRebT ZiriTadi mrudis M
wertils. Tu Wogrs mivcemT β -kuTxis biseqtrisis mimarTulebas da mas
gadavitanT zenitze, maSin Wogri miiRebs mimarTulebas Q wertilisaken da
Tu am mimarTulebiT gadavzomavT r-is mniSvnelobas, miviRebT Q wertils
(ZiriTadi mrudis Sua wertils).
amis Semdeg B wverodan A da C mimarTulebiT gadavzomavT BE
sigrZes. adgilze miviRebT E da E1 wertilebs (α kuTxis wveroebs), Semdeg
E da E1 wveroebidan orive mxareze gadavzomavT tangensebs da adgilze
miviRebT Sebrunebuli mrudeebis T1T2 da T3T4 wertilebs.
agreTve, E da E1 wertilebidan biseqtrisis mimarTulebiT
gadavzomavT Б -mniSvnelobas da miviRebT mrudis Sua 1Б da 2Б wertilebs.
amrigad, adgilze miviRebT serpantinis mTavar wertilebs, e.i. movaxdenT mis
dakvalvas.
14.6. barometruli niveloba
119922
T0 P0 Δ 0 υ0
T0 P Δ ii υ1
P T0 υ1
P T υ
rogorc adre aRvniSneT, niveloba aris moqmedebaTa erToblioba,
romlis saSualebiTac ganisazRvreba wertilTa Soris simaRleTa sxvaoba.
fizikuri xerxiT SegviZlia nebismier wertilze wnevis gansazRvra.
vinaidan, sxvadasxva simaRleze wneva sxvadasxvaa, amitom wnevaTa sxvaobebiT
SegviZlia simaRleTa sxvaobebis gamoTvla.
barometruli niveloba aris niveloba haeris wnevis gansazRvris
saSualebiT. imisaTvis, rom gamoviyvanoT formula simaRleTa sxvaobis
gamosaTvlelad, gavixsenoT sxvadasxva kanonebi fizikidan.
viciT, rom haeri warmoadgens sxvadasxva airebis erTobliobas, e.i.
airebi qimiurad ganzavebulebi ar arian, magram cal-calke TiTeuls
TavianTi wnevebi aqvT (parcialuri wnevebi), garda amisa, gvxvdeba
SemTxvevebi, rodesac temperatura ucvlelia, magram wneva icvleba, agreTve
piriqiTac. Cven unda gaviTvaliswinoT bunebis yovelgvari movlenebi.
I-daltonis kanoni. Tu gazebi qimiurad ganzavebulebi ar arian, maSin
saerTo wneva udris parcialur wnevaTa jams e.i.
P P P P Pn= + + + ⋅⋅ ⋅ +1 2 3 (14.6.1)
II-boil-mariotis kanoni. mudmivi temperaturis dros simkvrive
pirdapir proporciulia wnevisa da ukuproporciulia moculobis (es
xdeba idealur gazebSi). e.i. temperatura iyo T0 , darCa T0 , wneva iyo P0 ,
gaxda P, simkvrive iyo Δ 0 , gaxda Δ i , moculoba iyo υ0 , gaxda υ1 . boil-
mariotis kanonis Sesabamisad SegviZlia davweroT
0 0 1
0i
PP
υυ
Δ= =
Δ (14.6.2)
am proporciidan ganvsazRvravT υ1 . υ υ10
0= ⋅PP
(14.6.3)
me-3 formula warmoadgens daltonisa da boil-mariotis
gaerTianebul kanons.
III-gei-lusakis kanoni. mudmivi wnevis dros moculoba pirdapir
proporciulia temperaturisa.
kanonis Tanaxmad υ υ ε= +1 1( T) , (14.6.4)
sadac ε =1
273 aris gazebis gafarToebis koeficienti.
gavaerTianoT es kanonebi. SevitanoT υ1 -is mniSvneloba (14.6.3) formulidan
119933
(14.6.4) formulaSi. gveqneba, υ υ ε= +PP
T)00 1( , am tolobis orive mxare gavyoT
υ0 -ze miviRebT υυ
ε0
0 1= +PP
T)( aq (14.6.2) formulidan SevitanoT moculobebis
Sefardebis mniSvneloba, miviRebT:
ΔΔ
0 0 1= +PP
T)( ε aqedan Δ Δ=+
PP T0
01
1 ε (14.6.5)
(14.6.5) formula aris klapeiron-mendeleevis formula, romelic
aerTianebs samive kanons, e.i. amyarebs damokidebulebas simkvrivesa, wnevasa
da temperaturas Soris.
ganvixiloT 186-e naxazi.
nax. 186
Cvens mizans Seadgens h-is gansazRvra. formulis gamosayvanad unda
gavakeToT sami daSveba:
I-daSveba. TiTqos Q1 da Q2 wertilebi erT vertikalze mdebareoben
(aseTi SemTxveva bunebaSi aris ).
II-daSveba. TiTqos orive wertilze anaTvlis aReba erTsa da imave
dros xdeba (sinamdvileSi winaswar xdeba drois daweseba, vTqvaT 12 saaTze
aviRoT anaTvali).
III-daSveba. TiTqos wertilebze anaTvlebis aRebis dros wneva
mudmivia.
119944
muSaobis Tanmimdevroba: Q1 da Q2 wertilebSi vdgebiT barometrebiT
da viRebT anaTvlebs. barometrebi sindiyiania da anaTvlebi zevidan qveviT
izrdeba.
cxadia, anaTvali dabal Q1 wertilSi ufro didi iqneba, vidre maRal
Q2 wertilSi. anaTvalTa Soris sxvaoba B1–B2 aris sindiyis svetis sxvaoba,
gamowveuli wnevaTa sxvadasxvaobiT.
gavixsenoT ziarWurWlebis kanonic. viciT, rom ziarWurWlebSi
sxvadasxva simkvrivis mqone gazebisa Tu siTxeebis wonasworobis dros maTi
simaRleebi ise Seefardebian erTmaneTs, rogorc maTi simkvriveebi.
hB B
q
1 2−=
Δ, (14.6.6)
sadac q-aris sindiyis simkvrive, Δ -ki haeris simkvrive.
(14.6.6) formulidan ganvsazRvroT h. h q B B= −Δ
( )1 2 . Tu am formulaSi
SevitanT cnobil mniSvnelobebs q–sindiyis simkvrive =13.6; Δ -haeris
simkvrives (14.6.5) formulidan, miviRebT
hqPP
T)(B B= + −0
01 21
Δ( )ε , (14.6.7)
aq P0 – wnevaa zRvis doneze (sawyisi wneva) =760 mm
Δ 0 _ haeris simkvrivea zRvis doneze
q = 136.
P-ki udris 2
21 BBP += . (14.6.8)
barometriT muSaobis dros viRebT temperaturul anaTvlebsac T1 da
T2-s. saSualo temperatura TT T
=+1 2
2.
Tu (14.6.7) formulaSi SevitanT P-s mniSvnelobas, me-(14.6.8) formulidan
gveqneba
hqP
T)B BB B
= +−+
210
0
1 2
1 2Δ( ε . (14.6.9)
gamosaxulebaSi 2 0
0
qPΔ
yvela sidide cnobilia, amitom SegviZlia
SemovitanoT aRniSvna 2 0
0
qPK
Δ= , maSin (14.6.9) formula gadaiwereba ase:
119955
h K T)B BB B
= +−+
(1 1 2
1 2
ε (14.6.10)
es aris frangi fizikosis babines formula. (14.6.10) formulaSi SevitanoT
aRniSvna K T)B B
H(1 1
1 2
++
=ε Δ da ΔH vuwodoT simaRlis barometruli
safexuri. e.i. barometrze anaTvali 1mm-iT rom Seicvalos, wertilebs Soris
sxvaoba unda iyos ramdenime metri. praqtikulad igi daaxloebiT 10-11
metrs udris. e.i. 1mm-iT rom Seicvalos barometrze anaTvali 10-11 metriT
zeviT (an qveviT) unda avideT. (14.6.10) formula miiRebs saxes
h H B B= −Δ ( )1 2 (14.6.11)
arsebobs sxvadasxva cxrili maT gamosaTvlelad. axla (14.6.11) formulas
mivceT logariTmuli saxe. gavixsenoT maTematikidan mwkrivebis daSla.
λg xx
M x M x11
2 210 10+−
= ⋅ ≈ (dawvrilebiT n. TevzaZis „sainJinro geodeziis“ me-3 tomSi aris
ganxiluli).
vTqvaT XB BB B
=−+
1 2
1 2
, maSin λ λg
B BB BB BB B
gBB
MB BB B
1
12
1 2
1 2
1 2
1 2
1
210
1 2
1 2
+−+
−−+
= = ⋅−+
, aqedan
B BB B M
gBB
1 2
1 2 10
1
2
12
−+
= λ babines formulaSi (14.6.10)-Si SevitanoT es mniSvneloba,
miviRebT:
h KM
T) gBB
= +2
110
1
2
( ε λ .
am gamosaxulebaSi KM2 10
- cnobili sididea, amitom aRvniSnoT KM
N2 10
= ,
maSin
h N T) gBB
= + ⋅(1 1
2
ε λ , (14.6.12) es aris logariTmuli saxe.
am formulas aZleven agreTve cxrilur saxesac. amisaTvis gadavweroT
(14.6.12) formula ase
h N T TT
g gB g gB= +++
− + + −( ) ( )1 11
760 76000
1 2εεε
λ λ λ λ , (14.6.13)
119966
nax. 187
aq T0 -saS. temperaturaa, e.i. cnobili sididea. (14.6.13) formulidan amoviRoT
wevri 11 0
++
εε
TT
da gardavqmnaT ase
2 2 3 30 0 0
0 02 2 3 3 2 3 2 4 3
0 0 0 0 0 02 3 2
0 0 0 0
1 1(1 ) (1 )(1 )1 1
11 ( )( ) 1 ( ),
T T T T T TT T
T T T T TT TT TTT T T T T T
ε ε ε ε ε εε ε
ε ε ε ε ε ε εε ε ε γ
+= + = + − + − + ⋅⋅⋅ =
+ +
= − + − + − + − =
= + − + − = + −
sadac ( )ε ε ε− +20
30T T gamosaxulebaSi yvela sidide cnobilia, amitom
aRvniSnoT γ -Ti. miRebuli mniSvneloba SevitanoT (14.6.13) formulaSi.
miviRebT
h N T g gB g gB T T= + − + + − + −( )( )[ ( )]1 760 760 10 1 2 0ε γλ λ λ λ .
SemovitanoT aRniSvnebi: ( ) ( )H N T gB1 0
1
1 760= + ε λ ,
( ) ( )H N T gB2 0
2
1 760= + ε λ .
( )H1 aris I-wertilis miaxloebiTi altitudi, e.i. miaxloebiTi simaRle
zRvis donidan. (H2)-ki II-wertilisa. Tu SevitanT am aRniSvnebs, gveqneba
h H H T T H H T T H H T T= − + − = + ⋅ − − − ⋅ −[( ) ( )][ ( )] ( ) ( ) ( ) ( ) ( ) ( )2 1 0 2 2 0 1 1 01 γ γ γ ;
e.i. h H H H H T T= − + − −[( ) ( )] [( ) ( )]( )2 1 2 1 0γ (14.6.14)
(14) formula aris miaxloebiTi altitudebis saSualebiT wertilTa Soris
simaRleTa sxvaobis gamosaTvleli (aq T0-nebismieri daTqmuli ricxvia.
vTqvaT 15ο da misgan gamoiTvleba danarCenebi).
a) diferencialuri barometri
Tavis droze mende-
leevma mogvawoda diferen-
cialuri barometri Semdegi
konstruqciisa (nax. 187).
I wertilSi gavaRebT
onkans, haeri miawveba milSi
navTs da gaCerdeba raRac B1
119977
anaTvalze. Semdeg vketavT onkans. gadavitanT II wertilSi, anaTvali ukve
icvleba da xdeba B2 (II wertilSi onkans ar vaRebT).
sxvaoba B1-B2 Seadgens wnevaTa sxvaobas. mendeleevis barometri
imdenjer zustia sindiyis barometrze, ramdenjerac sindiyis simkvrive
metia navTobis simkvriveze. ЦНИГАИК-Si gaakeTes barometri, romelic
mosaxmareblad metad kargia, radgan moculobiT pataraa.
b) aneroidi
radgan sindiyiani barometri Zneli gamosayenebelia, amitom saWiro
Seiqmna usindiyo barometris Seqmna – mas aneroidi uwodes.
aneroidSi haeri amotumbulia, magram vinaidan absolutur amotumbvas
ver axerxeben, raRac narCeni haerisa mainc aris Sig, e.i. aris misi
temperaturac, rasac aCvenebs Sig aneroidSi moTavsebuli Termometri.
garda amisa drekadobis kanonis Tanaxmad, liTons aqvs Tviseba, romelic
proporciulad ar gadaadgilebs isars, maSin rodesac aneroidis Skalaze
danayofebi daniSnulia urTierT tolebi. e.i. rom miviRoT aneroidiT swori
anaTvali wnevisa, amisaTvis aneroidze anaTvalSi unda SevitanoT aRniSnuli
Sesworebebi.
vsargeblobT Semdegi empiriuli formuliT
B A bt C A a= + + − +( )760 (14.6.15)
am formulaSi A – aneroidze aRebuli anaTvalia;
bt – Sesworeba Siga temperaturisaTvis,
sadac t – Siga temperaturaa;
b – temperaturuli koeficienti;
C – Sesworeba skalisaTvis, skalis koeficienti;
a – aneroidis mdgomareobis Sesworeba.
(14.6.15) formuliT aneroidze aRebuli anaTvali gadagvyavs barometrze
aRebul anaTvalze. aRniSnuli Sesworebebidan yvelaze didia a –
Sesworeba.
rodesac, erTdroulad viRebT anaTvlebs barometrze da aneroidze,
anaTvalSi Segvaqvs Sesworebebi bt da C, garda amisa anaTvlebi mainc
119988
gansxvavdebian erTmaneTisagan da es gansxvaveba aris swored a -aneroidis
mdgomareobis Sesworeba.
vTqvaT, ara gvaqvs pasporti aneroidisaTvis da gvinda ganvsazRvroT
es Sesworebebi. amisaTvis viqceviT ase. oTaxSi (I-sarTulze) viRebT
anaTvals aneroidze, (haeris temperatura oTaxSi t1-ia), gamoviTvliT
B A bt C A a1 1 1 1760= + + − +( ) ,
Semdeg gamovdivarT gareT, gamovitanT aneroids, viRebT anaTvals
(haeris temperatura ukve t2-ia) gamoviTvliT.
B A bt C A a2 2 2 2760= + + − +( ) .
axla gamovakloT B2-s B1-i miviRebT
B B A A b t t C A A2 1 2 1 2 1 1 2− = − + − + −( ) ( ) ( ) (14.6.16)
am gamosaxulebaSi (A1-A2)-sxvaoba mxedvelobaSi ar aris misaRebi,
radganac simaRleTa sxvaoba oTaxsa da ezos Soris didi ar gveqneba.
amitom (14.6.16) gamosaxuleba gamartivdeba da miiRebs aseT saxes
bB B A A
t t=
− − −−
( ) ( )2 1 2 1
2 1
e.i. ganvsazRvravT b-s.
aviRoT aneroidze anaTvlebi iseTi pirobebisaTvis, rom ar gvqondes
temperaturuli sxvaoba, aramed gvqondes simaRleTa sxvaoba. maSin (t2 – t1) –
sxvaobas vugulvebelvyofT da imave (14.6.16) gamosaxulebidan gveqneba
CB B A A
A A=
− − −−
( ) ( )2 1 2 1
1 2
.��
Pa -s gansazRvrisaTvis viyenebT (14.6.15) formulas, saidanac
a B A bt C A= − + + −[ ( )]760 .
muSaobis wesi
aneroidiT muSaobis dros unda gvaxsovdes, rom ar SeiZleba
mcxunvare mzeSi anaTvlis aReba, unda davdgeT CrdilSi, an qolgis qveS,
gavCerdeT daaxloebiT naxevari saaTi da mxolod amis Semdeg SegviZlia
muSaobis dawyeba.
yovel sadgurze viRebT 5 cnobas: 1) daviWerT Surdula Termometrs
da daaxloebiT 2 wuTs vatrialebT, aviRebT anaTvals,� gavaCerebT cota
xans da kidev vatrialebT 1-wuTs, isev viRebT anaTvals. 2) aneroids
zevidan vartyamT xels, raTa gamovides inerciis mdgomareobidan, aviRebT
119999
anaTvals A-s. 3) aviRebT aneroidis temperaturas t-s. 4) aviRebT anaTvals
saaTze. 5) gavzomavT simaRles aneroididan miwamde. amiT mTavrdeba
sadgurze muSaoba.
radgan aneroidisaTvis cnobilia b,C da a vsazRvravT A, e.i.
SegviZlia gavigoT B (14.6.15) formulidan. agreTve viciT T-c, amitom
sabolood vmuSaobT (14.6.14) formuliT.
muSaobis meTodi
gazomvis warmoeba SegviZlia movaxdinoT erTi aneroidiTac da
oriTac, umjobesia ori aneroidiT muSaoba.
erTi aneroidiT muSaobis dros: pirvel wertilze viRebT anaTvals,
Semdeg mivdivarT da yvela sadgurze viRebT anaTvals. ukan dabrunebisasac
yovel sadgurze viRebT anaTvlebs, ra Tqma unda, sawyis I wertilSic
viRebT anaTvlebs. pirveli da bolo anaTvali gansxvavdebian
erTmaneTisagan, rac gamowveulia drois sxvadasxvaobiT. am gansxvavebas
anaTvlebs Soris Semdeg vanawilebT yoveli sadgurisaTvis Sebrunebuli
niSniT.
roca vmuSaobT ori aneroidiT: I sadgurze orive aneroidiT
erTdroulad viRebT anaTvlebs, Semdeg erTi rCeba amave I sadgurze, meore
teqnikosi ki meore aneroidiT midis win. winaswar vdebT pirobas, rom vTqvaT
yovel naxevar saaTSi aviRoT anaTvlebi da amitom yovel naxevar saaTSi
orive aneroidze aiReba anaTvlebi.
anaTvalTa damuSavebis dros erTmaneTs vadarebT erTsa da imave
saaTSi miRebul anaTvlebs.
ori aneroidiT muSaobiT ufro meti sizuste miiRweva.
Tanamedrove aneroidebSi miiReba sxvaoba 1-dan 4-metramde. e.i.
aneroidiT warmoebuli muSaoba sakmaod tlanqia, magram Cven viciT, rom
geodezistebi yovelTvis muSaoben aucilebeli da sakmarisi sizustiT. raime
sainJinro samuSaoebis Casatareblad, vTqvaT gzis gayvanis dros, nagebobis
aSenebis dros, xidebis mSeneblobis dros, saWiroa Catardes pirveladi
kvleva-Ziebani, romelic did sizustes ar moiTxoven. swored aseT
samuSaoTa Casatareblad ufro meti sizuste, vidre aneroidiT muSaobiT
miiRweva, saWiro ar aris.
220000
nax. 188
XV Tavi
farTobebis gamoTvla
sainJinro saqmeSi farTobebis gamoTvlas metad didi mniSvneloba
eniWeba. arsebobs farTobebis gamoTvlis mravali meTodi, romelTagan Cven
mxolod zogierTs SeviswavliT:
1. geometriuli xerxi;
2. grafikuli xerxi;
3. ujrediani qaRaldis xerxi;
4. awonvis xerxi;
5. koordinatebis (analizuri) xerxi;
6. meqanikuri xerxi (amsler-koradis planimetriT);
a) ganvixiloT farTobebis gamoTvla analizuri xerxiT, rodesac
saWiroa farTobis gamoTvla samkuTxedisaTvis.
samkuTxedis farTobi tolia, Tu
F bh=
2 (viciT h da b);
F bc A=2
sin (viciT b,c da A)
F b A CB
=⋅2
2sin sin
sin (viciT b,A da C, aq
B gamoTvlilia);
F P P a P b P c= − − −( )( )( ) (viciT marto gverdebi, sadac P a b c=
+ +2
).
Tu gamosaTvleli far-
Tobi SegviZlia warmovidgi-
noT samkuTxedebad (an cno-
bil geometriul figurebad),
gamoviTvliT calkeuli sam-
kuTxedebis farTobebs da
Semdeg maTi jami mogvcems
saZiebel farTobs.
farTobis gamoTvlaSi
is Secdomebi Seva, rac misi nax. 189
220011
nax. 190
nax. 191
elementebis gazomvis dros Sevidnen.
2. grafikuli xerxiT farTobebis gamoTvlis dros gamoiyeneba adre
Sedgenili gegmebi. gegmebidan amoiReben im elementebs, romlebic saWiroa
farTobebis gamosaTvlelad. grafikuli xerxi ufro tlanqia geometriul
xerxTan SedarebiT, radganac masSi garda im Secdomebisa, romlebic
analizuri xerxiT farTobebis gamoTvlaSi Sedis, emateba Secdomebi:
adgilze gazomili monacemebis naxazze datanisa, naxazis deformaciisa da
naxazidan grafikulad elementebis gansazRvrisa.
3. ujrediani qaRaldis xerxiT farTobebis gamoTvlis dros
gamoiyeneba gamWvirvale milimetrebiani qaRaldi, romelsac vafarebT
gamosaTvleli farTobis zedapiris gegmas.
vTqvaT, gegma Sedgenilia 1
5000
masStabSi e.i. viciT, rom
1sm2 – 2500kv.m = 14ha.
1mm2 – 25m2
daviTvliT ramdeni ujrediT
daifara, gamosaTvleli farTobi mTliani
ujredebiT da nawilobriv dafarul ujredebs ki vafasebT TvaliT.
4. awonvis xerxiT farTobis gamoTvla mdgomareobs SemdegSi: viRebT
gamWvirvale qaRalds (vaskovkas), movWriT gansazRvrul farTs, avwoniT
saafTiaqo sasworze da vTqvaT 100kv.sm gamovida 100 grami e.i. 1sm2=1gr.
axla gadaviRebT gegmas vaskovkaze, SemovWriT gansazRvrul farTs, avwoniT
da viTvliT saWiro farTs.
marTalia farTobebis gamoTvlis es
xerxi maRali sizustiT ar xasiaTdeba,
magram geologebisa da metyeveebisaTvis
misi sizuste sakmarisia.
5. koordinatebis xerxi (analizuri).
vTqvaT, gamosaTvlelia xuTkuTxedis
Fx x
y yx x
y yx x
y y
x xy y
x xy y
=+
− ++
− ++
− −
−+
− −+
−
1 22 1
2 33 2
3 44 3
1 55 1
5 44 5
2 2 2
2 2
( ) ( ) ( )
( ) ( )
220022
farTobi. Tu viciT misi yvela wveros koordinatebi, misi farTobi
SegviZlia warmovidginoT, rogorc sami trapeciis farTobis jams
gamoklebuli ori trapeciis farTobi
( , , ' , ' , , ' , ' , , ' , ' , , ' , ' )f. f. f. ,4,4',3'_f. f.1 2 2 1 2 3 3 2 3 1 5 5 1 5 4 4 5+ + −
es gamosaxuleba gavamartivoT Semdegnairad
2 1 2 2 3 3 4 4 5 5 1 1 2 2 3 3 4 4 5 5 1F x y x y x y x y x y y x y x y x y x y x= + + + + − − − − − ,
2 1 2 5 2 3 1 3 4 2 4 5 3 5 1 4F x y y x y y x y y x y y x y y= + − + − + − + −( ) ( ) ( ) ( ) ( )_ .
es gamosaxuleba mokled ase Caiwereba
F x y yi i ii
n
= −+ −=∑1
2 1 11
( ) ,
anda Tu davaproeqtebdiT X RerZze, maSin miviRebdiT
F y x xi i ii
n
= −+ −=∑1
2 1 11
( ) ,
anda ufro martivad dasamaxsovrebeli wesiT farTobi gamoiTvleba,
Tu amovwerT x-ebisa da y-ebis yvela mniSvnelobebs, Cveni aRebuli
magaliTisaTvis (xuTkuTxedisaTvis) gveqneba
x1, x2, x3, x4, x5, x1
y1, y2, y3, y4, y5, y1
da aviRebT namravlTa jams marcxnidan marjvniv mimarTulebebisaTvis (\)
da vaklebT namravlTa jams marjvnidan marcxniv mimarTulebebisaTvis (/),
samkuTxedisaTvis
x1, x2, x3, x1
y1, y2, y3, y1
7. farTobebis gamoTvla meqanikuri xerxiT (planimetris Teoria).
meqanikuri xerxiT farTobebis gamoTvlisaTvis vsargeblobT amsler-
koradis planimetriT.
planimetri Sesdgeba ori berketisagan: sapoluso da
Semomvlelisagan. Semomvlel berketze amTvleli meqanizmia. am or berkets
aqvT erTi saerTo wertili_saxsari.
planimetri qaRaldze sami wertiliT idgmeba. planimetrze aiReba
anaTvali 4 cifrisagan, pirveli cifri aiReba ciferblatze, Semdegi cifri
dolze, mesame cifri vernierze. ukanaskneli cifri aris SeTavsebuli Strixi
220033
nax. 192
vernierze, dolis sigrZis meaTasedebSi (dolis sigrZis 1
1000 aris
planimetris sizuste da aRiniSneba τ -Ti).
Sedegi miiReba Semdegnairad: aiReben sawyis wertilSi anaTvals,
Semoatrialeben wvetanas, mivlen isev sawyis wertilSi, aiReben anaTvals.
meore anaTvali minus pirveli anaTvali gvaZlevs saWiro cifrs n-s.
saWiroa P-farTobis gamoTvla (daStrixuli farTis gareSe), (nax.
267) jer gamovTvaloT elementaruli farTobi f. oa1b1b2a2o. am farTobis
gamoTvlis dros dauSveben Secdomas (daStrixul nawils), e.i. f.
oa b b a o oa b b b a o dP1 1 2 2 1 1 1 2 2≈ =f. ' -Ti avRniSnoT. naxazidan
dP a b b a oa a a b b R dh R d R d= + + = + + ⋅f. f. f.1 1 1 2 1 2 2 1 2 12
121
212
' ' α β ,
am formulaSi dh=rkalebiT aRniSnul manZilebis jams
dh d rd= +λ β ,
e.i. dP R d R d R d R rd= + + +12
12
112
12
λ α β β .
mTeli farTi ki udris elementaruli
farTobebis jams – integrals,
∫ ∫ ∫ ∫ ∫+++=P L
drRdRdRdRdP0 0
2
0
2
0
2
01
21
21 2
121 π π π
ββαλ .
maSasadame miviRebT
P R L R R R r= + + +12
12
12π( ) ,
π( )R R R r Q212
12+ + = -Ti avRniSnoT.
L-aris sigrZe, romelic gaiara dolma,
gamovsaxoT is τ -ebSi:
L n= ⋅τ , e.i. R L R n1 1= ⋅ ⋅τ , R C1 ⋅ =τ -Ti avRniSnoT, maSin 1R L Cn= -
Cn-aris planimetris safasuri, romelic Seesabameba planimetris
dolis sigrZis 1-meaTased farTobs.
maSasadame
P Cn Q= + (1) planimetris formula
axla ganvixiloT is SemTxveva,
rodesac gamosaTvleli farTobi ise
mcirea, rom planimetri Sig ar Caidgmeba,
O
r R1
a1
b1 R
nax. 193
220044
r
R1 R
R
b1
K O
nax. 267
R2
maSin planimetrs vaTavsebT farTobis gareT.
aq P Cn=
farTobi udris elementarul farTobTa jams (ar gveqneba Q-didi
wris farTi).
axla ganvsazRvroT Q-s geometriuli mniSvneloba.
naxazidan
R K r R22 2
12= + +( ) ,
sadac K R r2 2 2= − ,
SevitanoT K2-is mniSvneloba
R R r r R R r R R R r22 2 2 2
12
12
12
12 2= − + + + = + + ,
e.i. miviReT gamosaxuleba rac adre avRniSneT
Q-Ti e.i. Q R= ⋅π 22
Q-yofila didi wris farTobi, romlis radiusi
yofila R2-re.
maSasadame, planimetrma gamosaTvleli farTobi dayo didi wris
farTobebad da danarCeni farTobebi gardaqmna sworkuTxedebis jamad.
rodesac planimetri farTobis gareT aris moTavsebuli, maSin
farTobi gadaiqceva sworkuTxedebis jamad.
planimetriT rom vimuSaoT, saWiroa
C da Q codna.
C–s gamoTvla
cnobili r-radiusiT SemovxazoT wre
(patara), e.i. farTi iqneba πr P2 = ' (cnobi-
lia). planimetri moTavsebulia farTis
gareT, am SemTxvevaSi farTobi gamoiTvleba
ukve cnobili formuliT
P C n'= ⋅ , saidanac ucnobi wevri C Pn
=''.
P'
OO
rr RR11
aa11
bb11 RR
nax. 195
nax. 194
220055
nax. 196
P"
O
rR1
a1
b1 R
Q-s gamoTvla
aviRoT didi wre cnobili radiusiT. maSasadame misi farTobi cnobili
iqneba. cnobili formuliT vwerT
P Cn Q" "= + , aqedan Q P Cn= −" " .
aRniSnaven QC
q= -Ti. igi ganyenebuli
ricxvia da gviCvenebs, ramdeni safasuri mo-
Tavsdeba did wreSi. am aRniSvnis gamoye-
nebiT P C n q= +( ) , esec planimetris for-
mulaa.
planimetris Semowmeba
planimetrs aqvs 2-Semowmeba.
1) doli da vernieri erTmaneTisagan ar unda iyvnen arc Zalian Sors
da arc Zalian axlos, maT Soris papirosis qaRaldi unda gadiodes.
2) dolis sibrtye marTobi unda iyos Semomvleli berketis RerZisa.
Semowmeba xdeba Semdegnairad: roca polusi marjvnivaa, SemovuvliT
da aviRebT anaTvals n1, Semdeg berketi marcxnivaa, SemovuvliT_aviRebT
anaTvals n2, Tu piroba daculia, unda miviRoT erTi da igive anaTvali,
anda mcire sxvaobis dros viRebT saS. ariTmetikuls am ori anaTvlisas,
winaaRmdeg SemTxvevaSi planimetri moiTxovs Sesworebas.