Problemas Resueltos Cartograf­a Matemtica

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Descripciones y ejemplos basicas sobre cartografia en el uso de la geologia

Transcript of Problemas Resueltos Cartograf­a Matemtica

Ejercicio 1

1. Dadas las coordenadas UTM de dos bases de replanteo:

XYH(m)HusoElipsoide de HayforfdB1537874.934751640.93161.7929a6378388B246545.214729813.85294.1130e^20.00672267e'^20.0067681702Ko0.9996

hY las coord. geodsicas del puntoP42.5968222222-8.5284805556W209.49

a) Calcula la distancia UTM en el huso 29 desde las dos bases.

1. Primero calcularemos las coordenadas geodsicas del punto B2.

x-453636.244497799y4731706.53261304 = Sm

A1.0050739874m6367654.49129146'0.7430846851B0.0050846837n16107.0276636481' 10.7456046875C0.000010717p16.9744554954' 20.7456057596D0.0000000208q0.0219307596' 30.7456057642.7200632303sexa

Obtenemos n y N

NnTanCosB26388278.662656410.00365312260.92342199320.7346770793

Obtenemos las coordenadas geodsicas

B20.7432764516-0.0964365733B242.5866036862-5.5254086423B242.58660368620.4745913577Huso 29-8.5254086423Huso 300.74327645160.0082831818

2. Ahora obtenemos las coordenadas UTM de B2 y P en el huso 29

B242.58660368620.4745913577B20.74327645160.0082831818P42.59682222220.4715194444P0.74345479870.0082295668

A1.0050739874m6367654.49129146B0.0050846837n16107.0276636481Sm B24716880.5967474C0.000010717p16.9744554954Sm P4718015.75510313D0.0000000208q0.0219307596

Nn^2pXYB26388228.658704720.00366883496364876.97576413B2538943.3018781324715102.98950294P6388232.486611450.00366763216364888.4175133P538684.9110389954716236.28871472

3. Calculamos las distancias UTM entre los puntosXYHNn^2pAxAyDB1537874.934751640.93161.790.7490362946388352.342605090.00362997156365246.67827181D utm B2 a P-258.39083913741133.29921177681162.382mB2538943.3018781324715102.98950294294.11D utm B1 a P809.9810389945-35404.641285284435413.905mP538684.9110389954716236.28871472209.49

b) Acimut UTM y acimut geodsico.AxAyAcimutB1 a P809.9810389945-35404.64128528443.1187188229178.6894260407B2 a P-258.39083913741133.29921177686.0590184212347.1561835288

Calculamos la reduccion angular dTxyreducidas de translacion y escalaNmnmpmdTsegundosB137890.08603441384753542.34693878dT B1 a P6388292.414608270.00364880186365067.54789256-0.0000166805-3.4405950321B238958.88543230484716989.7854171dT B2 a P6388230.572658090.00366823356364882.696638720.00000054390.1121960601P38700.39119547274718123.53812997

Calculamos la convergencia de meridianos.

B10.0055146234B20.0056051386P0.0055699399

Obtenemos el acimut geodsico.

B1 a P3.1242167658179.004434966sexa179016B2 a P6.0646241037347.477365477sexa3472939

c) Distancia reducida al horizonte medio de B1 y B2

1. Calculamos el modulo de deformacion lineal puntual.

B10.9996176459B20.9996186569K1 B1 y B20.9996181514P0.9996184101

2. Clculo de la distancia sobre el elipsoide. (d2)3. Clculo de la distancia sobre el eliposide (cuerda). D1R6376665.58677967

D 2 B1 a P35427.433290426D1 B1 a P35427.3877265626D 2 B2 a P1162.8264603902D1 B2 a P1162.826458779

4. Distancia reducida al horizonte medio.

H medio227.95Dr B1 a P35428.6541679201 mDr B2 a P1162.8680269446 mKh = R0.9999642538

Ejercicio 2

2. Dadas las coordenadas geodsicas de los verticas y las alturas ortomtricas.

HElip. HayfordV141.93229263890.025414505.3a6378388V241.9071101389-0.32318675337.1e^20.00672267e'^20.0067681702Ko0.9996

a) Calcular las coordenadas UTM de los vrtices. (en los husos 31 y 30)

Huso 30HV10.73185656950.0528034355505.3A1.0050739874m6367654.49129146V20.73141705190.0467192047337.1B0.0050846837n16107.0276636481C0.000010717p16.9744554954Sm V14644198.6321635D0.0000000208q0.0219307596Sm V24641401.48491338

Nn^2pXYHusoV16387983.826666350.00374577166335686.17161378V1750847.1399982794646768.8312574430V26387974.415547220.00374872916335647.54911531V2722028.4126954494643010.4516531330

Huso 31HV10.7318565695-0.0519163196505.3A1.0050739874m6367654.49129146V20.7314170519-0.0580005504337.1B0.0050846837n16107.0276636481C0.000010717p16.9744554954Sm V14644198.6321635D0.0000000208q0.0219307596Sm V24641401.48491338

Nn^2pXYHusoV16387983.826666350.00374577166335680.24714517V1253367.5917230264646621.2237536631V26387974.415547220.00374872916335722.8883144V2224352.1850288334644887.4029195431

b) Acimut de V1 a V2

Huso 30AxAyAcimutAcimut UTM-28818.72730283-3758.37960431074.5827063971262.5697353004xyV1-246731.1007172614648480.61600006Huso 31AxAyAcimutV2-275758.1182184554646746.10136009Acimut UTM-29015.4066941935-1733.82083411244.6527047712266.5803467088

Calculamos el acimut geodsico.

dT0.000005515-0.0347106329

Acimut geodsico4.6179996533264.5918899251

c) Distancia UTM.

AxAyDD UTM-29015.4066941935-1733.820834112429067.1629216588m

d) Distancia sobre el elipsoide.

1. Calculamos el coeficiente de demormacion lineal puntual K1

K1 V11.000348338D229054.3340672703R6361786.64623558K1 V21.000534756Km1.000441547D129054.3088171568

e) Distancia terreno reducida al horizonte medio.

Hm421.2

kh = R V10.9999205789kh = R V20.9999470145Dr29056.2324344038kh = R Vm0.9999337967

Ejercicio 3

3. Desde el vrtice V1 se ha realizado una observacion a una base, obtenidose:

HElipsoideV140.40833333334.0252777778508a6378388e0.00672267DgAcimutVime'0.00676817022890.25108.3631591.08361.561.3ko0.9996

a) Calcular las coordenadas UTM del vrtice.

Huso 31HV10.70525846190.017894473508A1.0050739874m6367654.49129146B0.0050846837n16107.0276636481C0.000010717p16.9744554954Sm V14474947.00099086D0.0000000208q0.0219307596

Nn^2pXYHusoV16387416.241790430.00392416496362448.94301452V1586998.5705639034473661.6321427730

b) Coordenadas UTM de la base (obtener la convergencia de meridianos y el acimut a la base).

1. Hallamos K1 y la convergencia de meridianos2. Hallamos la distancia sobre la cuerda del elipsoide D1

K1 V10.999693156AH-53.8479122037mH454.1520877963R6374920.36939861 V10.0116004723radD12889.5302843046

3. Hallamos D24. Hallamos la distancia UTM

D22889.5303090401D utm2888.6436740816

5. Hallamos el acimut cartogrfico.

Acimut1.879693283107.6984918962sexa

6 Obtenemos las coordenadas UTM del punto.

XYHBase589750.4932408844472783.46139871454.1520877963

Ejercicio 4

4. Dadas las coordenadas UTM y alturas ortomtricas de los vrtices.

XYHHusoElipsoideV1733480.094739868.923299.130a6378388V2257091.744729254.093348.231e0.00672267e'0.0067681702ko0.9996

a) Calcular las coordenadas geodsicas de los vrtices.

xy = SmV1233573.5194077634741765.6262505V2-243005.4621848744731146.54861945A1.0050739874m6367654.49129146B0.0050846837n16107.0276636481'0.74466440240.7429967432C0.000010717p16.9744554954' 10.74718508430.745516707D0.0000000208q0.0219307596' 20.74718611650.7455177813' 30.74718611690.7455177817Nn^2pV16388312.599959670.0036424596365127.88230328TanCosV26388276.773634340.00365371626365020.79399482V10.92635421490.7336040214V20.92325900830.7347367624

HusoV10.74656519760.0498095685V142.7750349534-0.1461219476300.0498095685V20.7448479788-0.0517388921V242.67664556380.035579843731

b) Acimut UTM de V1 a V2

1. Calculamos las coordenadas UTM de V2 en el huso 30.husoA1.0050739874m6367654.49129146V20.74484797880.05298086330B0.0050846837n16107.0276636481C0.000010717p16.9744554954Sm4726883.25344349D0.0000000208q0.0219307596

Nn^2pXYHusoV26388262.392922420.00365823496364977.80901109V2748739.6162452944729461.2588211730

2. Calculamos el acimut UTM

AXAYAcimutV1 a V215259.5262452944-10407.66117883012.1693690399124.2956901919

c) Calculamos el acimut geodsico.

dT -0.0000306671xyNnpV1233573.5194077634741765.62625056388312.599959670.0036424596365127.88230328 V10.033841993V2248839.1519060574731353.800341316388276.773634340.00365371626365020.793994826388294.686797010.00364808766365074.33814905

Acimut geodsico2.2031803657126.2329364628

d) Coeficiente de anamorfosis en V2

K V21.0003610628K1 V11.0002705382Km1.0003158005

e) Distancia UTM entre V1 y V2.

AXAYDD UTM15259.5262452944-10407.661178830118470.857m

f) Distancia reducida al horizonte

D218465.0255769009R6376673.94301022

D118465.0191255587H V13299.1Kh0.9994828976D r18474.6434599588mH V23348.20.9994752057Hm3323.650.9994790517

Ejercicio 5

5. Dadas las coordenadas geodsicas del punto inicial y final de un tramo de obra lineal.

ElipsoideI40.40.1Ea6378388F40.4-0.2We0.00672267e'0.0067681702ko0.9996

a) Calcular las coordenadas UTM de ambos puntos.

Huso 30V10.70511301780.0541052068A1.0050739874m6367654.49129146V20.70511301780.0488692191B0.0050846837n16107.0276636481C0.000010717p16.9744554954Sm V14474021.62097493D0.0000000208q0.0219307596Sm V24474021.62097493

Nn^2pXYHusoV16387413.150436550.00392513666335695.04634318V1763097.2751347624476847.4337483530V26387413.150436550.00392513666335660.65951526V2737632.8005399254475996.9280092530

Huso 31V10.7051130178-0.0506145483A1.0050739874m6367654.49129146V20.7051130178-0.0558505361B0.0050846837n16107.0276636481C0.000010717p16.9744554954Sm V14474021.62097493D0.0000000208q0.0219307596Sm V24474021.62097493

Nn^2pXYHusoV16387413.150436550.00392513666335671.73455582V1253879.1633740744476270.7988373231V26387413.150436550.00392513666335707.28281305V2228414.3142403934477150.2022231431

b) Distancia UTM entre ambos puntos.

Huso 31AxAyDV1 a V2-25464.8491336812879.403385820825480.0292722779

c) Acimut Utm y geodsico.

Huso 31AxAyAcimutxxNn^2pV1 a V2-25464.8491336812879.40338582084.746909273271.977867076V1-246219.3243556684478062.023646786387413.150436550.00392513666335671.73455582V2-271694.3635050094478941.778934726387413.150436550.00392513666335707.28281305dT-0.00000277956387413.150436550.00392513666335689.50868443

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