Formulas y relaciones entre sistemas de coordenadas en Geodesia Geometrica I | Esquemas y mapas conceptuales de Matemáticas

UNIVERSIDAD MAYOR DE SAN ANDRES FACULTAD DE TECNOLOGIA

CARRERA DE GEODESIA, TOPOGRAFIA Y GEOMATICA

GEODESIA GEOMETRICA I - GTG 352

AUX. DE DOCENCIA – UNIV. GROVER LOPEZ LAURA

-ϕ -λ -70° -69° -68° -67° -66° -65° -64° -63° -62° -61° -60° -59° -58° -57°

-9° 26.41 24.36 22.61 20.33 17.3 13.78 12.03 10.99 7.72 4.11 0.95 -2.53 -5.35 -7.7

-10° 27.4 26.28 24.8 23.44 21.39 18.49 15.91 14.46 11.13 7.23 3.32 -0.81 -4 -3.94

-11° 28.39 27.69 27.15 26.51 24.44 20.9 19.4 17.87 14.75 11.05 6.78 2.28 -1.91 -4.42

-12° 27.47 28.82 29.32 29.83 27 22.97 20.83 19.41 18.37 15.25 10.84 9.14 8.13 2.02

-13° 31.04 30.17 30.55 31.13 29.19 26.84 23.9 20.61 18.69 17.04 13.39 11.52 11.07 4.69

-14° 44.16 33.72 31.14 28.83 29.35 28.38 24.81 21.13 18.81 17.64 14.91 12.28 10.17 4.1

-15° 46.58 44.52 32.93 27.3 26.51 26.04 25.32 22.41 19.98 19.12 16.4 12.63 7.98 3.17

-16° 45.46 45.61 41 31.68 24.89 22.16 22.31 23.9 22.42 21.07 17.03 13.26 8.64 4.91

-17° 44.5 44.56 44.38 42.72 36.12 24.88 21.08 22.6 22.27 21.64 19.1 14.39 10.41 7.12

-18° 38.3 44.4 44.31 45.04 41.42 37.53 30.4 21.61 21.6 21.86 20.66 15.89 13.27 9.92

-19° 35.43 43.51 43.17 44.06 43.29 39.15 30.25 21.99 23.06 21.51 18.9 17.33 15.74 12.49

-20° 33.58 41.78 43.2 43.17 43.6 41.77 29.76 23.16 23.58 22.12 19.71 17.7 16.49 13.89

-21° 31.62 37.89 42.78 42.39 42.92 39.66 29.39 22.41 22.81 20.96 20.9 18.21 15.88 13.86

-22° 33.31 36.83 43.26 43.57 42 38.89 27.49 23.22 21.41 20.52 20.18 17.75 15.54 14.11

-23° 33.28 36.31 40.28 43.77 43.43 37.45 24.72 23.54 22.58 21.08 19.47 17.82 15.91 13.02

MODELO GEOIDAL SOFTWARE NIMA CALCULATOR EGM-96 (1°x1°)

WS 290° 291° 292° 293° 294° 295° 296° 297° 298° 299° 300° 301° 302° 303°

-ϕ -λ -70° -69° -68° -67° -66° -65° -64° -63° -62° -61° -60° -59° -58° -57°

-9° 29,80 27,03 24,26 21,49 18,72 15,95 13,18 10,41 7,64 4,87 2,10 0,00 -2,10 -4,20

-10° 32,00 29,20 26,40 23,60 20,80 18,00 15,20 12,40 9,60 6,80 4 1,80 -0,40 -2,60

-11° 32,30 29,63 26,96 24,29 21,62 18,95 16,28 13,61 10,94 8,27 5,60 3,37 1,14 -1,09

-12° 32,60 30,06 27,52 24,98 22,44 19,90 17,36 14,82 12,28 9,74 7,20 4,94 2,68 0,42

-13° 32,90 30,49 28,08 25,67 23,26 20,85 18,44 16,03 13,62 11,21 8,80 6,51 4,22 1,93

-14° 33,20 30,92 28,64 26,36 24,08 21,80 19,52 17,24 14,96 12,68 10,40 8,08 5,76 3,44

-15° 33,50 31,35 29,20 27,05 24,90 22,75 20,60 18,45 16,30 14,15 12,00 9,65 7,30 4,95

-16° 33,80 31,78 29,76 27,74 25,72 23,70 21,68 19,66 17,64 15,62 13,60 11,22 8,84 6,46

-17° 34,10 32,21 30,32 28,43 26,54 24,65 22,76 20,87 18,98 17,09 15,20 12,79 10,38 7,97

-18° 34,40 32,64 30,88 29,12 27,36 25,60 23,84 22,08 20,32 18,56 16,80 14,36 11,92 9,48

-19° 34,70 33,07 31,44 29,81 28,18 26,55 24,92 23,29 21,66 20,03 18,40 15,93 13,46 10,99

-20° 35 33,50 32,00 30,50 29,00 27,50 26,00 24,50 23,00 21,50 20 17,50 15,00 12,50

-21° 34,70 33,19 31,68 30,17 28,66 27,15 25,64 24,13 22,62 21,11 19,60 17,23 14,86 12,49

-22° 34,40 32,88 31,36 29,84 28,32 26,80 25,28 23,76 22,24 20,72 19,20 16,96 14,72 12,48

-23° 34,10 32,57 31,04 29,51 27,98 26,45 24,92 23,39 21,86 20,33 18,80 16,69 14,58 12,47

MODELO GEOIDAL PARA BOLIVIA WGS-84 (1°x1°)

FORMULARIO GEODESIA GEOMETRICA I – GTG 352

MODELOS GEOIDALES

-ϕ -λ -70° -69° -68° -67° -66° -65° -64° -63° -62° -61° -60° -59° -58° -57°

-9° 27.0298 25.1073 23.2003 20.3444 17.2667 14.1916 12.643 11.7749 8.6822 5.3406 2.6722 -0.7563 -4.4381 -7.1344

-10° 28.1835 27.2091 25.4591 24.0078 22.1821 19.2941 16.5253 14.7266 11.261 7.3659 3.855 -0.0698 -3.6979 -4.0796

-11° 29.5022 28.6315 28.1321 27.7936 26.4275 22.7759 20.586 17.9321 14.1412 10.2055 6.2845 1.9727 -2.4458 -5.1721

-12° 28.3937 29.2906 29.5266 30.3629 28.3669 24.7991 22.4496 20.1884 17.9292 15.069 10.1941 6.8603 6.5279 0.3516

-13° 30.8851 30.6362 29.4114 30.0065 28.4058 26.8849 25.0983 21.7839 19.1533 17.0473 12.7828 9.0631 9.1254 2.0939

-14° 46.3069 36.6681 29.4837 25.7999 26.7059 26.6824 24.6079 21.7212 19.5301 18.1004 14.2885 12.4127 9.6969 3.742

-15° 48.8031 47.1353 32.2687 23.5936 23.2256 23.8217 24.3611 22.6666 20.5177 19.77 16.9471 12.5904 8.3811 3.096

-16° 45.2273 45.8079 40.7347 31.7689 23.0713 21.1896 22.264 24.6901 23.8515 21.9529 18.0348 14.2779 9.883 5.4959

-17° 44.1901 44.1713 46.2639 45.3404 38.4434 23.8836 21.7979 23.0539 22.7301 21.4376 19.2606 15.6104 12.2152 9.2579

-18° 37.9299 44.5828 45.3873 46.4398 43.3974 38.0414 31.1484 21.7248 21.0576 20.7656 19.3527 16.6227 14.2788 11.0005

-19° 36.0901 43.7186 43.3866 44.2565 44.5102 40.8299 30.6723 22.1835 22.8043 20.5342 17.8395 16.9445 14.8474 12.0801

-20° 34.9615 42.2982 42.7658 42.8259 42.9305 41.8907 29.6731 22.927 23.7491 22.4529 19.5796 17.3665 15.6443 13.4263

-21° 32.9047 38.6352 42.5286 42.3575 42.3333 38.5161 28.5408 22.8468 23.6251 21.643 21.0624 18.1236 15.1246 13.2116

-22° 35.4321 37.7872 42.7067 43.498 42.2002 39.2432 28.0211 23.7073 22.631 20.3452 20.3527 17.968 15.3766 14.1621

-23° 32.9047 36.3117 38.9886 43.0851 42.5289 38.2307 24.7708 23.6967 23.5192 21.7802 20.0318 17.5768 15.6625 13.1121

MODELO GEOIDAL SOFTWARE ALLTRANS CALCULATOR EGM-08 (1°x1°)

Formulas y relaciones entre sistemas de coordenadas en Geodesia Geometrica I, Esquemas y mapas conceptuales de Matemáticas

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CARRERA DE GEODESIA, TOPOGRAFIA Y GEOMATICA

WS 290° 291° 292° 293° 294° 295° 296° 297° 298° 299° 300° 301° 302° 303°

MODELOS GEOIDALES

h=Altura elipsoidal

H=Altura Ortometrica

N=Separacion del geoide (altura geoidal) Calculo auxiliar para hallar h (altura)

Radios De Curvatura

√( 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑)

𝑎 ( 1 − 𝑒^2 )

( 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑)

𝑀 𝑠𝑖𝑛^2 𝛼 + 𝑁 𝑐𝑜𝑠^2 𝛼

𝑁^2 − 𝑎^2

𝑁^2 𝑒^2

𝑀^2 /^3 − (𝑎( 1 − 𝑒^2 ))^2 /^3

𝑀^2 /^3 𝑒^2

𝑅 𝑒^2

Longitud de Arco Meridiano 𝑺𝑴 (Desarrollada y no desarrollada)

Donde: A.B.C.D.E son valores que dependen de la excentricidad ‘’e’’ 𝑆𝑀 =^ 𝑎^ (^1 −^ 𝑒

( 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑)

Longitud de Arco paralelo Exceso esferico

√( 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑)

]

Area de Trapecio Elipdoidal

Donde: A’.B’.C’.D’.E’ son valores que dependen de la excentricidad ‘’e’’ 𝑆𝐴 =^

𝜋 ∗ 𝑏^2 ∗ ∆𝜆

( 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑)^2

𝜋 ∗ 𝐾 ∗ 𝑏^2 ∗ ∆𝜆

Variables para Long. De Arco Meridiano y Areas

Sistemas De Referencias

Problemas Geodesicos Puisant

Problema Directo Problema Inverso

√ 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑𝑜

𝑠𝑖𝑛 𝜑 1 = 𝑠𝑖𝑛 𝜑 0 ∗ 𝑐𝑜𝑠 Ŝ +cos 𝜑 0 ∗ 𝑠𝑖𝑛 Ŝ ∗ 𝑐𝑜𝑠 𝛼𝑃 0 −𝑃 1

𝑎 ( 1 − 𝑒^2 )

( 1 − 𝑒^2 ∗ 𝑠𝑖𝑛^2 𝜑𝑚)

√ 1 − 𝑒^2 ∗ 𝑠𝑖𝑛^2 𝜑 1

𝑎 ( 1 − 𝑒^2 )

( 1 − 𝑒^2 ∗ 𝑠𝑖𝑛^2 𝜑𝑚)

√ 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑𝑜

𝑐𝑜𝑠 Ŝ = 𝑠𝑖𝑛 𝜑 0 ∗ 𝑠𝑖𝑛 𝜑 1 + cos 𝜑 0 ∗ 𝑐𝑜𝑠 𝜑 1 ∗ 𝑐𝑜𝑠 ∆𝜆

Exceso esferico

𝑒′′^ =

𝑎 1 ∗ 𝑏 1 ∗ 𝑠𝑖𝑛 ∢𝐶 1 ∗ ( 1 − 𝑒^2 𝑠𝑖𝑛^2 𝜑)^2

2 ∗ 𝑎^2 ∗ ( 1 − 𝑒^2 )^ ∗ 𝑠𝑖𝑛 (00°00′01′′)

Donde:

𝑒^2 = 𝑝𝑟𝑖𝑚𝑒𝑟𝑎 𝑒𝑥𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑑𝑎𝑑

a b

f

a

a b

e

a

e = f (2 − f )

e

e

e

a b

e

b

b = a (1 − e )

e

e

e