referen cn plochy sou radnicov e syst emy na referen cn ch
TRANSCRIPT
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Obsah
1 Referencnı plochyElipsoidKouleRovina
2 Souradnicove systemy na referencnıch plochachSouradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnicePravouhle a polarnı souradnice
3 Dulezite krivkyOrtodromaLoxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Obsah
1 Referencnı plochyElipsoidKouleRovina
2 Souradnicove systemy na referencnıch plochachSouradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnicePravouhle a polarnı souradnice
3 Dulezite krivkyOrtodromaLoxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Referencnı plocha
Definice:Matematicky definovana plocha, ktera nahrazuje zemske nebo jinevesmırne teleso nebo jeho cast, urcena pro geodeticke akartograficke vypocty (definice VUGTK).
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Elipsoid: je prostorove teleso tvorene mnozinou vsech bodu, jejichzpoloha vuci zadanemu bodu (stredu) splnuje podmınky danenasledujıcı nerovnicı.
x2
a2+
y2
b2+
z2
c2≤ 1
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Elipsoid trojosy:
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Rotacnı elipsoid (sferoid)
x2
a2+
y2
b2+
z2
c2= 1→ a = c
Rotacnı elipsoid je urcen dvema konstantami:
a - hlavnı poloosa
b - vedlejsı poloosa elipsoidu
Dalsı parametry elipsoidu:
e - numericka vystrednost (prvnı excentricita) e2 = a2−b2
a2
i - zplostenı elipsoidu - i = a−ba
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Rotacnı elipsoid (sferoid)
x2
a2+
y2
b2+
z2
c2= 1→ a = c
Rotacnı elipsoid je urcen dvema konstantami:
a - hlavnı poloosa
b - vedlejsı poloosa elipsoidu
Dalsı parametry elipsoidu:
e - numericka vystrednost (prvnı excentricita) e2 = a2−b2
a2
i - zplostenı elipsoidu - i = a−ba
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Rotacnı elipsoid (sferoid)
x2
a2+
y2
b2+
z2
c2= 1→ a = c
Rotacnı elipsoid je urcen dvema konstantami:
a - hlavnı poloosa
b - vedlejsı poloosa elipsoidu
Dalsı parametry elipsoidu:
e - numericka vystrednost (prvnı excentricita) e2 = a2−b2
a2
i - zplostenı elipsoidu - i = a−ba
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Elipsoid rotacnı:
Obem:4
3πabcJan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Druhy eliposidu:
jestlize a > b > c , jde o obecny (trojosy) elipsoid
jestlize a > b = c , jde o protahly sferoid
jestlize a = b > c , jde o zplostely (diskovity) sferoid
jestlize a = b = c , jde o kouli
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Elipsoid trojosy (3:2:1):
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Priklady pouzıvanych elipsouidu
a [m] b [m] e
Zachuv el. 6 376 045 6 355 477 0,0802571310
Besseluv el. (r. 1841) 6 377 397 6 356 078 0,081 696
Hayforduv el. (r. 1909) 6 378 388 6 356 911 0,081 991 889
Krasovskeho el. (r. 1940) 6 378 245 6 356 863 0,081 813 333
IAG (r. 1967) 6 378 160 6 356 774 0,081 820 565
WGS-84(r. 1984) 6 378 137 6 356 752 0,081 191 910
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
http://cs.wikipedia.org/wiki/Elipsoid
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Elipsoid a hlavnı polomery krivosti:http://gis.zcu.cz/studium/mk2/multimedialni_texty/
M =a(1− e2)
(1− e2 sin2 ϕ)32
N =a
(1− e2 sin2 ϕ)12
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Diferencialy uhloveho a polednıkoveho oblouku na elipsoidu:
dsp = M · dϕ
dsr = N · cos(ϕ)dλ
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Referencnı koule
x2 + y2 + z2 = r2
Pomocı koule lze snadneji resit ulohy kartografie a geodezie.
Casto pouzıvame kouli jako mezistupen prizobrazenı elipsoidudo roviny.
Vyuzitı pro konstukci map malych merıtek
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Referencnı koule
x2 + y2 + z2 = r2
Pomocı koule lze snadneji resit ulohy kartografie a geodezie.
Casto pouzıvame kouli jako mezistupen prizobrazenı elipsoidudo roviny.
Vyuzitı pro konstukci map malych merıtek
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Volba polomeru: Polomer koule jako tzv. strednı polomer krivosti:
Rm =√
(M · N) =a√
1− e2
1− e2 · sinϕ
Pro CR se strednı geodetickou sırkou ϕ = 49o30′ a pouzitıBesselova elipsoidu je Rm = 6380703.6105m.
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Volba polomeru:
Koule o stejnem obemu:
R = (a2b)1/3
Koule se stenym povrchem
Pro CR se strednı geodetickou sırkou ϕ = 49o30′ a pouzitıBesselova elipsoidu je Rm = 6380703.6105m.
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Diferencialy uhloveho a polednıkoveho oblouku na kouli:
dsp = R · dU
dsr = R · cos(U)dV
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Referencnı rovina
Tecna rovine ve zvolenem bode
Pouzitı pro mala uzemı (20 x 20 km).
Pro vetsı uzemı znacne vyskove a polohove odchylky.
Nulova krivost, nebere v potaz zakrivenı Zeme.
Mapy velkych merıtek: statnı mapove dılo.
Nelze pouzıt pro mapy malych a strednıch merıtek, velkezkreslenı
Tecna rovina v bode, princip azimutalnıch zobrazenı.
Matematicka kartografie - prevod elisposidu resp. koule do roviny
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
ElipsoidKouleRovina
Oprava delek pri nahrade tecnou rovinou:
10 km - 0.3 cm
25 km - 3.5 cm
50 km - 25.8 cm
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Obsah
1 Referencnı plochyElipsoidKouleRovina
2 Souradnicove systemy na referencnıch plochachSouradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnicePravouhle a polarnı souradnice
3 Dulezite krivkyOrtodromaLoxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Souradnice na elipsoidu:
Zemepisne souradnice ϕ, λ
Geocentricka sırka β
Redukovana sırka Ψ
Pravouhle prostorove souradnice X,Y,Z
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Zemepisne souradnice (Geodetic coordinates)
Zemepisna sırka ϕ - uhel, ktery svıra normala elipsoidu srovinou rovnıku.
Zemepisna delka λ - uhel, ktery svıra rovina polednıku srovinou nulteho polednıku
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Geocentricka a redukovana sırka
Geocentricka sırka β: Uhel spojnice stredu elipsoidu se bodemna elipsoidu s rovinou rovnıku
Redukovana sırka Ψ: Uhel spojnice prumetu bodu lezıcıho naoskulacnı kruznici s rovinou rovnıku.
Souradnice bodu:
x = a cos Ψ
y = b sin Ψ
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Geocentricka a zemepisna sırka
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Geocentricka a zemepisna sırka
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Kartograficke souradnice
Pro optimalnı volbu zobrazenı a s tım souvisejıcı polohouzobrazovacı plochy je vhodne urcovat polohu bodu pomocı tzv.kartografickych souradnic. Osa zobrazovacı plochy jiz nebudetotozna s osou zemskou a definujeme novy kartograficky systemsouradnic. Prusecık osy plochy s referencnı plochou jekartografickym polem Q. Definice kartografickych souradnic S(kartograficka sırka) a D (kartograficka delka) je pak analogicka ksouradnicım zemepisnym U, V.
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Kartograficke souradnice
http://gis.zcu.cz/studium/mk2/multimedialni_texty/
index.html
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
Souradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnice
Izometricke souradnice
ds2 = f (ξ, η) · (ξ2 + η2)
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
OrtodromaLoxodroma
Obsah
1 Referencnı plochyElipsoidKouleRovina
2 Souradnicove systemy na referencnıch plochachSouradnice na elipsoiduZemepisne souradniceKartograficke souradniceIzometricke (symetricke) souradnicePravouhle a polarnı souradnice
3 Dulezite krivkyOrtodromaLoxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
OrtodromaLoxodroma
Ortodroma - definice
Ortodroma (recky orthos - prımy, dromos - cesta) je nejkratsıspojnice dvou bodu na kulove plose (napr. povrchu Zeme). Tvorıji kratsı oblouk hlavnı kruznice (jejı stred splyva se stredem Zeme).
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
OrtodromaLoxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
OrtodromaLoxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
OrtodromaLoxodroma
Loxodroma - krivka na referencnı plose, ktera protına vsechnypolednıky pod stale stejnym uhlem - azimutem.
tanA =R · cosU · dV
R · dUPo integraci:
tanA =U2 − U1
arctanh(sinV2)− arctanh(sinV1)
Delka loxodromy:
s = RV2 − V1
cosA
http://cs.wikipedia.org/wiki/Loxodroma
Jan Jezek
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Referencnı plochySouradnicove systemy na referencnıch plochach
Dulezite krivky
OrtodromaLoxodroma
Jan Jezek