topografie_probleme rezolvate

FACULTATEA DE IMBUNATATIRl FUNCIARE SI INGINERlA MEDIULUI DISCIPLINA TOPOGRAFIE AN II

PROBLEME PROPUSE SPRE REZOLVARE

Responsabil disciplina Conf Manea Raluca

1

4

EXEMPLU NUMERIC INTERSECTIA INAINTE Se dan coordonatele punctelor 1234 Se cere sa secalcuIeze coor donat I eel nnctulUl P

peT x y

1 819916 459 797 2 786278 843262 3 506365 807199 4 533996 462201 P 678157 637705

EXEMPLU NUMERIC INTERSECTIA INAINTE

Se dau coordonatele punctelor 123 Se cere sa se calculeze coordonatele punctului P

peT x y

1 750238 133972 2 433084 1058435 3 401849 1637839 P 532795 1285629

2

2

3

EXEMPLU NUMERIC INTERSECTIA INAPOI - COLLINS Se dau coordonatele punctelor 123 si directiile unghiulare Se cere sa se calculeze coordonatele punctelor C si P

PCT X Y 1 706559 378390 2 698259 502572 3 719022 565 939 C 763402 511658 P 557952 483006

PS PV Directii unghiu1are orizontale 1 753380

P 2 1232083 3 1446581

1

p 3

EXEMPLU NUMERIC INTERSECTIA INAINTE PCT X y

A 2899138 2894056 B 2919771 4288426 C 1784974 4711902 D 1630229 3358846 P 2342056 3710021

NT unghi

Valoare unghi

NT unghi

Valoare unghi

1 300780 5 39 5556 2 49 0200 6 63 5739 3 727761 7 515294 4 449557 8 39 5114

A

D

8 r

1

6

(

2

c

4

TRANSMITEREA LA SOL A PUNCTELOR STATIONABILE

1Egt28851gA B

2

peT x y

A 1341713 2255 838 P 1274685 2482735 B 1345992 2716 766

5

2

INTERSECTIA INAPOI -REZOLVAREA HANSEN

p

peT x y

1 1265787 3618466 2 1353465 4045428

Dp Q 20194 m

6

INTERSECTIA LINIARA

B

2

peT x y A 1691797 3818 234 B 1815746 4224657

DAB 424 90m DA 1 16719m DB] 33498m DB2 16194m DA2 32084m D12 29849m

7

INTERSECTIA INAPOI - REZOLVAREA CASSINI ~

MARTINIAN

Se dau coordonatele punctelor 123 si directiile unghiulare orizontale Se cere sa se calculeze coordonatele punctului P

peT x y 1 1440 4360 2 1490 4510 3 1400 4650

PS PV Directii unghiulare orizontale 1 1832475

P 2 248 8433 3 3096358

8

TRANSMITEREA LA SOL DIN PUNCT NESTATIONABIL

p

N

3

2

peT x y M 1810 461 0 P 1850 4740 N 1820 4920 D12 14560M D23 14560m

9

INTERSECTIA INAINTE

1

4 Se dau coordonatele pun ctelor 12 34 in sistem Stereografic 1970 Se cere sa se calculeze coordonatele punctului P prin metod a intersectiei inainte din toate cele 4 combinatii

NRPCT X(m) Y(m) 1 1384626 1351344 2 1428085 1359953 3 1432488 1418500 4 1370629 1404885 P 1409032 1394105

10

INTERSECTIA INAPOI - REZOLVAREA COLLINS

c

2 3

p Se dau coordonatele punctelor 1 2 3 in sistem Stereografic 1970 Se cere sa se calculeze coordonatele punctului P prin metoda intersectiei inapoi - rezolvarea COLLINS

NRPCT X(m) Y(m) 1 1307346 1380 634 2 1289709 1431389 3 1297135 1470 967 C 1330981 1427490 P 12323 93 1436804

11

INTERSECTIA INAPOI - REZOLVAREA HANSEN

B

A

p Se dau coordonatele punctelor A B in sistem Stereografic 1970 Se cere sa se calculeze coordonatele punctelor P si Q prin metoda

rsectiei mapOl - rezo varea inte 1 HANSEN NRPCT X(m) Y(m)

A 1373554 1347892 B 1405718 1470796 p 1341927 1388144 Q 1354256 1436447

Dpo 4985m

Denumire Valoare unghi (grade unghi centesimale)

UI 1157569

Uz 259357

U3 295698

U4 1238088

12

- -- - -- -- - --- -~_ _----------

INTERSECTIA LINIARA

1

A

B

2 Se dau distantele dintre puncte si coordonatele punctelor A si B Se cere sa se calculeze coordonatele punctelor 1 si 2 prin metoda intersectiei liniare

NRPCT X(m) Y(m) A 1402485 1365618 B 1353144 14688 27

DAB 11440m 10209m 6701m 7006m 7136m 10437m

DB2

DA2

DA 1

DB l

D 12

13

--

CO-pI ~~o b lE Kf Ro V~ pound

S9R~ ~~ 2 0L V A-Re

- lNT ER s ~c middot shy

gtrH)~~Q A eY eMLU NUMt R iC iNrE R ~ e CTiA lUA-iJJTE

Se olOAM c-e~f)Moplusmn~ ~d~ A)2 1~ )4

Se ~ do ~ ~tM~L~Q t ()~fMo~ ~~~ ~

r(T X(MA) Y( v)

amp~~ ~~~ ~5~ 1 ~ t

1

~ ~ ~ Q-~ -

~4~ 2G~

3 50( 6~5 go~ 4ltJ3

~ 5~) ltlt3C 4~1 104

~ G~~ ~5+ Cbi- q-05 fj

I

~t A L--shy -GA Lt-- -- shy -BAP

-----shy I ----r---shy ~

--~)

--- e--~~

3 -ampb4

f~~ ~~ei~ m~vbei lM~ANk

~ ~~-kJl~

- - - - -

A middot ll~~~V OluQ f J -t~cQ

CAJ ~~~~~~ Iii -k~-~ b ~ ~ Ceamp~J ~M~ku4t U~~olJL

ltX~ z ~+ ~ 16 middotf v ~ exJ gtG 3~ c 8~ )

rb~ == 2- 0 - (cx~ + ()(t) z 20 0 - t3 G5fgs == A2~ 3421

2 M~ ~ et~Q~ 1 )A1

DAl V(X~ - x~) ~ +- (YI - Y1)l = V(BA9 g~G -Hbn~f+45Lfst- - 84 J~ l) L

) J)~1 ~ J 1Lt tlG SJJt1 =) J)~l = ~ g 4 g lt~ t55 ~ ~ g ~ q3BMshy

t ]) ~F gtg4 3~~~

2gt Apt~OJUQ -klt2-~~~ LM- bLUvlt~ A t l ~Qjwt~

QqQ~ dlticIA tQerL J)AP ~ J)2P middot

J)~l _ ])AP _ ]) ~

YM-~~ I)ll IX1 1JJJt ()(~

_ J)_-- IJ- cx_ _ _ r~g ~ AA 3~ 3~ cg _ 221 Al_1t__ l ~ g~ ( 4B J) ~p = - _ 0 ~r

1 ~ l6 ~ ~A1G Jlt2~

D 1)1l ~ 0( z ~~ 4 Sbg j-~~+-1G ( O z ~r~2 15R A4t

-Igt ~jbl~~1~H~

~ Coi Q-~-~kiJUlampl

iJ ~ i 1 I amp4 middot~ 161 middot-45S q-g1shy ---Ut L=lCgtJlL~ ~~_Y~ 0= OJe~ 1-~(1lamp _ ampA~J 4G shytr

A1 ~ + otc ~ AfL~S~~G ~ -Oo-- OJC- k~ A~ 3 ~ ~ 1~-

~l ltL00~- Si4 41q~9 A06 5101~

~~P = ~1- + 0( ~ A06 ~tol ~ t 6+ 2-bl O A[fl g3-1 J

-amp1 -= -G--A - cXl -= O 5 ampl-ol~ - bC-1 middot g~ 2 ( 9 4+3 ~ ~

6 C tAamp e-G O-d~cdJampt lAIy d-uJuL ~

i ~ -= Y A +1l QQIIB-4 ampt ~~~ 1- Jlt 4 coo 141 gL

X~ ~~ltj lt3~b - A ~t ~5~ ~ G i-~ A5i- ~

Ipi y ~ +J)4P ~~ ~ ~ ~ 65 i--lt3+ +0-1-4 )t- ~~-- ~ io g311 ~ pr~ 459i-~ rt- + Ait-l ~ 0 == ( 61- ~O 5 IIushy

x~ ~ l + l) eo~-B--~ ~ q- G1i- + 101 ~ LeJ lG9 f fgt4

XyI - l~b 1-t-2-- A0 ~ ~ 0~ zz G~2 ~ Cit- ~

YpIi s YJ- +J)JV ~M -E1-1f = ~4 ~ 1~ 1 +1gt 1 1~ I ~ 1b9ft34 y tc g4~ 1Gl ~ 105551- G61- CKJ5 ~

V~ ampfttLJt cJt L~~amp1A edd9 1 p~+ ~ ~ l1uM~~ t middot~ lP

Xp = Grt A5middot+ - f J cvu ~-tQ II

Yp~ ( ~ t Q- 05 ~

b) P e l~vC-Al~ ~~~tAQ ~Nl ~~ui 11 1J~

A 01 MM~tv(~ ~t~~ c4

(Xb ~ ~ AOG4~) 6(4 = 5t ql~ J ~

i I~J- 2 00 ~ - ()(b + ()(It )= 100 ~ - AAR144+- z g1 l 55~ ~

2 ecJ~J ~Ovtthi middotD 1~

] 1) = f 01--Ky + l Yl--YJgt t c Vrt~cng - 5 ()b5~ + - shy

])J-~~ vt +- ~~ 1l~ - ~O~ ~ ~1 )t +( 4~ lbl- --~ Oi- A93) 1shy

J)l z ~- i-S 5~ ~ffi4 J 1 11 ~t M~

3 A-pUQiJlQ -ttw-~ ~~~ ~ ~M~ 1 13~ ~~Lu

~)M~WlCL- cU0Vk~ )~ ~ 1)1gt bull

) l~ - J)-p _ ]) ~p shy

JtM ri ~M tX ~ I)(R-~

])~~ ~ ~th laquo4 - 1g1 121 ~~- 6t tlg~ = 12gt2 15g~ bj -= 0

jitv 1- ~~ ~A 1 pound5 5) 1

]))~ ~M ot~ d ~ J 111- gtl-~G~ 0l0 4 z 14( 33J AMshy~p z

Itv 1b1 if111t- g( 1amp55~ = ===

~ CcJ~ ~Qtvd-CAl~ c

- = ouck fb - V1- ~ M e 4-~ gO- ~ gq - 41J 1GZ ~1gt d i0J __ X- ~ 50( ~G 5 -- i-gG1i-g

t l-~ = =- OtC ~ OA lamp ~~ zz 100 + ampA ~1-O z 2-0 A 5~o ~

~l~ = -ampJ-~ -- ltXb- ~O ~ j5t-O +G ~ O~ G 4 1~ 9 ~ ~3 4 shy

-G = 400 - ( ()(~ - -B-~L) ~ 400 - 49 5t-1 15tO 41 El ~

5 tiliM te-~tlC~ p~ltJ-~ ~

fy ~ J- +]) Q troVl = t~~ -1~ +1~1 ltJ 2 ~coo2 G 9 A tJ~ Xp = t~~ 11-amp- A() ~ ~ klA CC 1-25tAM

Yf i - J- -f- J) l v ~ lP ~ amp~~ ~ G1 + 0-110g~ Ijj~ lG 9 A1- t Yfi ~ 8 ~ 1~1- ~o6 5)+ ~ G middot~+ +05 Mv

l ~+ ])1gtP ~ ~e-- tz 60 6 ~ ltj +1 ~t ~b~~ ~ t~ 35041g+ 50( ~~0 + AkA -ltj1 ~ ~k~ ~ f5+ ~

11gt +lJ~ riM ~bf -z ~ O1- Aj~ t 14~ b~ r ~ ~ ~50 41if go+ A ~~ - l~g ~ Cj4 G3Cf- 105 ~

middot~MoQQ Q ~~~t pe+rp

X -== Gt-~ middot ~ C)i- -shy~ - (01 5 v0shy

c) ljL~vOJUCt -bi~GA-G~~ AM tA1u~ut A ~4 p

A ccJL~ VIIA~tt fat ~tl~~

cX~- -f5 It o~ ~ J 0( G 1 g gG 5(~

~~ = 100 ~ C(C5 f amp6 ) = LOO- 9 4 10 4 x ~o5 middot1-93(2 ~

~ Ccamptului- cli~i J) ~it

J~ ~ (C 5()( ~C5 -6J~ 99Y+( ampCl-i9g - lrG2 1(1)

J)~4 z ~ 1-pound ~ ~11 + ~ sQ--~ G10 == 1 Irt ~ -Bi- Og J

])~ 4=- ~~~ ~ lcj AM

3 Cc~~ clt~t J)3 ~ J)4P oUlh-~1-i1laquo(~ l iM b 3~P

]) ~ 4 _ J)~ _ n ltP -

~~) ~ 6(C ~ eX5

_ ~LC Ao~ 9W-4U b5~ z 24t 3341amplt M1Vv AOS 11~c

Lf QoQw l) tyenUQlvvtDvC ~l

B ~ QIU-k bull 4- J = Mt-l 4G210 - Q~ Aq~ --Jt ltI- ~ _ Ib J1JJ Ej 3G - DOC JCIj

~~Lr -z i Cvcl-~ kl It~5~ = 400- 9~ H11 ~ ~o5og middott-~ ~

-t--~ = ~~lt + 0lt5= ~50 4l~ ~

it = ~4J - ltXc z )G --l11 shy

5~o1t~ tampel~~ ~ ~

Xf = X20 +])o Q~yB-b tz 50G 3bb +- 24L 6J If e-fX 36( 4l8b

X j= ~0GJ) (5 +At-L L = ~ +-~ ~ 5i- AM

f = + ])bY ~~~gt gO+ Agg +l4t gt31 1)1~ 3vo_418G If = gOt- ~~S- AG9 lt~4~ Qb f-fotJ AM

-r = XLt +) fp yen-~ G-4P -= 6~ ~ S)~G +~lt tlA )t- ~~ CS-~ 1)1l

l-- I~ 5) b (j~6 4- A44~(b ~ ~ ~+~ A51- AMrYplJ = ~4+ )~ 1)(M-44 P ~ 4G~ loA + ~lq- AQ~ oG~~tl1~ ftJ~ Lt~ 1 ~()A + A ~r) 50~ z ~ ~1- gt ~5 ~

J~ ~~ ~ ~~c4~~ ~

Xp = ~l A5+AM 1 foJL~ ~ p = ~ J ~ 1-0 ~ AM- e=======

amp) ~~VltMtGL ~~~~~cii- 1M 1lu~~JtAX ~ ~ 4~

A QoJL~ ~JJilG( Q-Itt ~amp~

C(i- ~G t-o1-5~ (X ~ = 5b Gb~G ~

bit 0-co-( IXt- + ampJ) =gCGOg3L

2 Qcamp~~ ~ JJ~4

)14 ~ V(~Aq~~b -5~~~ltj~)~ +( ~tj3 i-~)i- - 4~1 )01) A

1)~ 4 V~ A It-~b o-SCl ) DA4 n~ qlo 1M nm

~ ~ J)1 P i ])4P ttampte-lMQ 11~~

~ ~WJt 144P ~~~ _ )~r _ JJljp

M ~ 4 - ~ IXt- IM- 1(8

J) - ])~4~ ~lgti- _ 1g5q 1)4~ 5G t5t~ I) 1lf 1 g1P - z ~ 1 vUt - ~rv~br

c

~~~ ~O ~g

11 - D~4 1- ~M OltK - 2- amp5~ b l ~ S( c J~G - l~f 124 4p - tM- ~4 - 4t~ g~ G0 gg - t L-l

4 e~ GtlVvvt~~t

-BA = OJlt- k + 14v4 plusmn0Jk 4--a Ooo~ ~og 100- 0-n3m ~ - 1 es~ 10 _d

=) 444== A~~ lt Ltb41shy

-G-~p = ~A~ - lXR z A41 ~J ~ ~ ~

lt f -0 -6LIt~ t ()( l- zz 6G11-12 ~

5 CcQ~ C-l9ampl ~l PC+~

x~ z ~ +JJ ~~ ~ QQ~-G-A P ~ ~A~ ltjb + ~2t- Ltg ~ coo A41 ~ A~ Y-r = ~~ CjAG - ~ 4 A 1-5g ~+g ~6+ ~ pi - ~ +])~t IfMAt-~ p lt 59+~t- + 11i- it k ~J1IV ~ 42 ~ pi z 45 ~ ~gtt+ Alt-~O~ ~ ~ ~ q 1- GAM

l f ll

=- ~ 4 +])4P - C-~ e-Lt p ~ 5~~ ltj ~G -I- i1+ A~ - ~ 5b 211l i p

ll 5J~ Cj~~ + A44 ~~ ~ ~ +R Af5 1-- AM

ff II ~ Y4 + ])4P ~IIJ -G-~ p ~ -1 b1lfJA +llf- Amiddot11 ~ ~ 2-Q2shy

r tl 4Cl loA 1- kJj 5 ~ ~ _~~~ 106fM

rj)J ~eampt t~et~ Al r f)12 Jampl-- e-a-Q wQOlt ect

~ct~ OJL ~d1 Q(i Ak middot(aulbRe LSamp~ (9~~k et

lX2--t- cl) AM- ~wf~ 1 AL(p

~ Q~ ~k fwU1) c ~c~cpound etf u tvv+~ ~vcW loowi~te Q ~ vUJIM- ~ ~ ~ A11M ~ (1L MMSk Ot-v~

~ p = G31--fo~5 ~ G1t ~ WC ~

) ~ x p G1-amp C5t1 VOJVUOVW+ct 4 1 = G~ ~~b ~ middot

1 ~1~CviQ OJVkh-tt Q~

ct) ll e~fOAtQ CVA~QQ ~tVv -kAA~vvl~d A klP

A ccJL~ uN~~UvOhL 1l~~

OltJ - bq- 1~ A0 ~) ()(1 z b~ ~ Cj C

~~ = 1 0 - l cx ~ +oc~l) = A2b b 4 1 2~

~ ~ C~LuiuJ ~r~l~t

-GJ == ov~k E431Gt - ~5 5+~ + z plusmn oJlJ) -rw AA ~~ ~ i-G Al --~ +~ ~ l V _ ~ ~ ~ - -0shy

-9 = 100- ~t Lti~ 2 = AO~ 1) tOl~t t ~AP =- -G-~ 1- + (Xtt - A41 ~ b A 1d -Gt ~ lA - Oltl z 2b ~ Alb4~

~ Sc- ~ Qcueplusmnli~ cLru-pJvampampC A ~ 1f L -G - ~p -~ I ( - gt - Y1shy -~ A~ -- J ) 1(~ ~t~ -

A - I q- Ap - A-1

])tM ~~ ~~ ~ fiL -so QN1 I middot~Ov )JM t1A ~UA- Jt rkgtua ~~ ltUL I ~ ~-

~ fnW-1vW)U1--t ~~ ~ If ~~ Q~ ~~IM- ~b~ ~fu1e

middot = x~ b -G-~ p - X1~-e-~tgt T YQ - y~ f f1 +n ~

~~t~ ~ - -d11-a

Yvz lX p- X4) k~1r +~ iJow

Yp o (Xp- XL) ~-fl + It

X - ~4b 1Gl - 4 63~- +ampA(L~lt k ~ A ~l ~~n - ~ G) ) ~~ b1A~

~ ~ A41 ~J~l - ~ ~3 A1~1t X p b ~~ 4~5 - A01ltJ Oot - 4494 R5t - 2A40 3amp~ ANIo

- ~ JJ~ SO G~ A5t - A lt10~( ~~~~ - ~ ~ t5 lt2 ~~4 4X ~

~ Xp= ltt- A ~+AM

~D i z (~~~ k5~ ~A~ ~ ~~ QA ~ ~ ~~~1 - A4A TI~ ~ t lJ5ooJ 5 T _

-J

4- -1 59 1-~ ) j

+ 46~ 1--~ t- ~)

~) Ypi- A~+ 90 ~ +455 7-9i- G11-oG MA-

Yf 11 s ( ~q A)1- - l-2G11-amp) ~ Q~~ A~~4 +~~~ 2) ~ ~

p1~ == l- A bkl~ )(+A90 AA l5 ~1~) +- ~ 4J 1(2 z

~pi1- -aos 651--+ R4b -l~ 1 ~ 3 + +05 ~

~) Yp= ~0t lrw5 5 ~~ ~ Ggt t WGIvk

) 1e= GJi- ~ Of vA r

l~ ~~ a-l r~ ~ ~~ ru ~vQ t~ ~ UA JA- http

if( z G amp A~+ tv-

Ir Gt~+ tOG~

~ GUc lU-middot M ~u

U~let(l~ Ov~t~ cvri-~

evWNwk ~

x z G+~ middot A61- ~

f =- G0+ 5 MA

J~ 1 Xp -= ~~ A~t-~

~y = ~bt w6 M

j~o ~ t

p~ G+t ~ Sf-AM

~r== ~ Jl+ 1D ~~

1 O-JWv~ Y XfJ G1-~ A5t traquo 1p CD 3-i- ~OG AM

Jov~~JCt t5 Xp G~ A51-M lp~ Gbi- ~6 1M

I

Y1 ~etQ Q~ Vv Q-B~Q Q u1n Cl~el ~~ c QACJLQj- ~Clt-

9-olGllt ~veamp ~ ~MQ~ pJv

XY = 6~~ A~ 1Nvt SoL lIT +)rJItLmiddotA E~Te ~utJ A- ~ ( ~p - G31- ta5 ~

l9~Dl ~ peok ~~~ ~ -~~ VMJt ~ LoouAQVt~ pel P (JvJJc ~tl- W- ~ oJ~At te~ ~ ~tWbpound ~~

l

- -

se d0UL cewLwcJ~ ~u~m ) ) 3

Se- en~ 0lti ~ cot~~ e9ampwtltteet~ ~UM-Ju~ p

~cr X Y ~ +5028 ~~ ~~ +1 L 4 ~ ~ - O amp ~ ~058 ~35

3 4o g43 ~6~+ B~LJ

P 5~1 l95 A185 blamp

- - --_shy

f~ t Pilj~LcAtut i~~ ~ ~MwMke

CJi- 1~ -likkM-~ uJ0 ~ l~ (~M~W~)

~~(L buca-ampA~tkA4eQ

(1) ~1 ~Ct ~IA IllUflA~ ~ 1~

A CcamptraquoiuQ )JjAG~~ amptL~~

6( ~O 6 A Q bull I 1 Q ) _gt ~~ 100 ~ - ( ~1 1 8 o ( +coV58~~ )

CX1-~ ~ 1 1 O~ ~ Jj -~ A O-r A~ ~ L

I- ~ CoQtuhamp diJ~ J)~

])~ ~ if ( 4 () ~ 8 ~ ~ middot- t50 1j~ )t+ ( ~b ~t Rj~ middot- ~ ~ ~g 1-10) 1

-6 XII

J)~~ lt ~ (-~ 2 3 igt9)L+ ~g W1 t b Y~ A~

-[ -1) 3-0-r])-r~--45-~ ~

6 1pQltcVllC ~~ I)J1Wwtui ~tt j~+~ A~tgt FJv1LAt o~~

cU~ (ft- ]) ~ P ~ J)p

J~ J =])~ 9 = ) ~fl -=) JJ r = ]) ~ ~ 4i~ ()41 4

oraquo ~~ fd M ex1 1t1A 0( ~ tJ-tM 0~ ) - ]) -=- J) ~ ~ ~ At IX1

]) == 468 51 lv ~l lampDl 114 Ott~ 3 IM IIgt

A fdM- ~o+ ~18 ~ 4

]) ~ 4586 lt41- GOi5g~~ = ~qS -GG AAshy

I middot f~ M 0+ r~B~

4 Coiuiliamp ~vI-t~M~

( OJtt1qb Yo = OVlt~ +1 ~ H11 = plusmn M t bl 0 0551) A~ ~b Y- q -~ltg~~~ _ (- f) t

~~ -G ~~ = 100- 415 os ~ g ) -f~ A54 ~4 OJ ~ ~-Y3 ~ 354 9LtOl ~

-z z ---cJA ~ T ~ zz 1 middot3 1 ( 6 middot r~ G J ~ 6 LA 5 ]110 j

5 CcJL t-campzJ ( amp01d tYV1 cv-k~ i-vltampttJ-~ =v 1Kp = ~ 4 +])1Qro~ P - 150 l~E + jJ--4 OlA ~ (~j l A5 5 ~J() 5 3 1 1-9y~

~p ~ ~ ~ +D~ r l)fM- -f1 P~ ~3 j9 +l O-+ 2 2Lt 001tl middot iJL lA S5 Ll0 = II )~ g5 Gl 1AA

XI)ll ~ Xv +])~P ( 00 -tf~p ltOLg~ g+ 8i-5i-(G coo 1~ GGoAz 5Jl i-3Lf AA1shy

y1 YJ ~ ])3P I)j ~P ~ 4G11-amp J3+ 6t51G ~ 2J)) GbVA z 1JJ 5( 2 ~ iIA

A ~Uvt oJ~QdOJl ~CJvl~ ~~ OL camp ampl clamp1tt~ ~duk P Goc

Vl t cl l~ ~~vJctt( QQ AMi-Lt CJgtlt cLlt9-UR ~d4 ~ eo1~ )~~

Kp ~ 661l--3 ~ 11Mshy

fp~ A1~ 5 c l 1- MAshy

6) ~~l~u~Q -Jlt~t~ e-tL ~~ l h ht~wvurl~d 3QP Ceamp~ U+uru~ or4~~o1t l

6() = A ~ 1b A 2 ~ J - clt z l~) t5~) - gt rl 100 - l ~ 3 1) ~ g+1 91-5 t ) )

it ~ ~ ~= 1 54 0103~

1 e~~ c~~~ ]) ~~ ( ]) ~L)

])~~_ = y-( ~ J~-084- ~ 01 843) L + (1 0 5amp it 35- Co ~1- gJ3) L

----- b x -------- -- A Y ------shyj l ---~=--_ --shyg l _

)~1 y(+- 31 1) 5) L +(- 5t~ lt 04t ~ (33GG84 G2 oLr 4iA1shy

)])L 5 ~ 0 1~ iJ L

----- -------shy

- - - ~ l- --- 3P J

rWAM~ ~ ~-

bull )

]) ~ L ]) ~ ~ - - =

raquoraquo ~ L IJZ~ ex Lr

~ l i culuQ ampU~~9i

-vJ1 =- OlcC~ =-Jt ~ -t o4 oJLd~ ~ ~ 5~ 5g 2J ~O-J ~1g CL -+-jA 135 middott

-B-J-J z ) bl - lOO~ ~ ob ~U ( ~

-Gp ~-B3 l -l tX~ = ~ 0 1~ lg G +A ~ l~A g =- ~ 11(b 043

-G-l-~~ 4J-~ - 0(4 = AO~ Ll g G -- 13i-5l j - q-1 Gmiddott--11 J

is (0~ cM ~ Q ampL~~ ~dvlul P

Xp l

sz ~ J +])3i C lt9J -Gp ~ 401middot3 ~ q +~ni-G4~ t Glt) 612- Gro OY - 532=t g 5t~

YP 11gt +1)~ r ~M--tf3P ~ Go~ 8J9 +)1-5 16 y tdfyen- 312 bG O~ = ~ is5G~ 9~

Xp = Xl +])l P cro~l~~ ~ ~j o g~ t 14QAA2yen Q$J i-) ~ -tf~~ 5~U95~

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EXEMPLU NUMERIC INTERSECTIA INAINTE Se dan coordonatele punctelor 1234 Se cere sa secalcuIeze coor donat I eel nnctulUl P

peT x y

1 819916 459 797 2 786278 843262 3 506365 807199 4 533996 462201 P 678157 637705



peT x y

1 750238 133972 2 433084 1058435 3 401849 1637839 P 532795 1285629

2

2

3


PCT X Y 1 706559 378390 2 698259 502572 3 719022 565 939 C 763402 511658 P 557952 483006


P 2 1232083 3 1446581

1

p 3


A 2899138 2894056 B 2919771 4288426 C 1784974 4711902 D 1630229 3358846 P 2342056 3710021

NT unghi

Valoare unghi

NT unghi

Valoare unghi

1 300780 5 39 5556 2 49 0200 6 63 5739 3 727761 7 515294 4 449557 8 39 5114

A

D

8 r

1

6

(

2

c

4


1Egt28851gA B

2

peT x y

A 1341713 2255 838 P 1274685 2482735 B 1345992 2716 766

5

2


p

peT x y

1 1265787 3618466 2 1353465 4045428

Dp Q 20194 m

6

INTERSECTIA LINIARA

B

2

peT x y A 1691797 3818 234 B 1815746 4224657


7


MARTINIAN


peT x y 1 1440 4360 2 1490 4510 3 1400 4650


P 2 248 8433 3 3096358

8


p

N

3

2

peT x y M 1810 461 0 P 1850 4740 N 1820 4920 D12 14560M D23 14560m

9

INTERSECTIA INAINTE

1


NRPCT X(m) Y(m) 1 1384626 1351344 2 1428085 1359953 3 1432488 1418500 4 1370629 1404885 P 1409032 1394105

10


c

2 3


NRPCT X(m) Y(m) 1 1307346 1380 634 2 1289709 1431389 3 1297135 1470 967 C 1330981 1427490 P 12323 93 1436804

11


B

A



A 1373554 1347892 B 1405718 1470796 p 1341927 1388144 Q 1354256 1436447

Dpo 4985m


UI 1157569

Uz 259357

U3 295698

U4 1238088

12

- -- - -- -- - --- -~_ _----------

INTERSECTIA LINIARA

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Problema 8

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Problema 8pdf

Problema 8

Problema 8-4

Problema 5pdf

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INTERSECTIA LINIARA

B

2

peT x y A 1691797 3818 234 B 1815746 4224657


7


MARTINIAN


peT x y 1 1440 4360 2 1490 4510 3 1400 4650


P 2 248 8433 3 3096358

8


p

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3

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peT x y M 1810 461 0 P 1850 4740 N 1820 4920 D12 14560M D23 14560m

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INTERSECTIA INAINTE

1


NRPCT X(m) Y(m) 1 1384626 1351344 2 1428085 1359953 3 1432488 1418500 4 1370629 1404885 P 1409032 1394105

10


c

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NRPCT X(m) Y(m) 1 1307346 1380 634 2 1289709 1431389 3 1297135 1470 967 C 1330981 1427490 P 12323 93 1436804

11


B

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A 1373554 1347892 B 1405718 1470796 p 1341927 1388144 Q 1354256 1436447

Dpo 4985m


UI 1157569

Uz 259357

U3 295698

U4 1238088

12

- -- - -- -- - --- -~_ _----------

INTERSECTIA LINIARA

1

A

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NRPCT X(m) Y(m) A 1402485 1365618 B 1353144 14688 27

DAB 11440m 10209m 6701m 7006m 7136m 10437m

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A 1373554 1347892 B 1405718 1470796 p 1341927 1388144 Q 1354256 1436447

Dpo 4985m


UI 1157569

Uz 259357

U3 295698

U4 1238088

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Problema 8-4

INTERSECTIA INAINTE

1


NRPCT X(m) Y(m) 1 1384626 1351344 2 1428085 1359953 3 1432488 1418500 4 1370629 1404885 P 1409032 1394105

10


c

2 3


NRPCT X(m) Y(m) 1 1307346 1380 634 2 1289709 1431389 3 1297135 1470 967 C 1330981 1427490 P 12323 93 1436804

11


B

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A 1373554 1347892 B 1405718 1470796 p 1341927 1388144 Q 1354256 1436447

Dpo 4985m


UI 1157569

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- -- - -- -- - --- -~_ _----------

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1

A

B


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DAB 11440m 10209m 6701m 7006m 7136m 10437m

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Problema 8

Problema 8-4

Problema 5pdf

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scan0003

scan0004

Problema 5pdf

scan0001

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Problema 8pdf

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c

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11


B

A



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1

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