Guia Nº8

Profesor: Matas Saavedra A. Ayudante: Andrs Romn E.

GUA N 8 GEODESIA I

Estacin Punto visado

Angulo horizontal Angulo vertical Altura de la seal

Distancia inclinada

A hi =

1,40m

C B

00 00 00 37 34 53,10

90 41 30,20 ------------

1,25m ------------

17893,074m ------------

B hi =

1,45m

A C

00 00 00 47 48 24,00

------------ 89 53 37,00

---------- 1,34m

----------- -----------

C hi =

1,46m

B A

00 00 00 94 36 42,00

90 13 51,00 90 41 30,20

1,34m 0,00m

14728,70m ------------

mHmH BA

A

A

89,3720 10,3958

''55,25 '32 70 ''37,12 '37 70

''32,8 '18 31 ''65,29 '30 31

B

B

==-=-=-=-=

lljj

Calcule las coordenadas geogrficas de C y su cota.

Elipsoide de referencia : Internacional 1924, 2971 ; 388.378.6 == fma


Solucin: Dibujo de la poligonal B

mH B 89,3720

''55,25 '32 70

'32,8 '18 31

B

B

=-=-=

lj

C

A

10,3958

''37,12 '37 70

''65,29 '30 31

mH A

A

A

=-=-=

lj

Cierre angular de la poligonal

''3,0 3

''9,0 ''9,0 '0 0 +=+=+= unitarioErrorError

.estaciones de numero n""con ; ''46,33''2n''2

estacin ''1orden I

=

El error de cierre angular se encuentra dentro la tolerancia por lo tanto se puede compensar.

180 ''10,59 '59 179 '42,30' 36' 94:C ''3,0 '42,00' 36' 94:C

'24,30' 48' 47: '0,3' '24,00' 48' 47:

''40,53 '34 37: ''3,0 ''10,53 '34 37:

=S=S+++

BB

AA

''10,59 '59 179 '42,00' 36' 94:C

'24,00' 48' 47:

''10,53 '34 37:

=S

B

A


Calculo de la superficie de la poligonal Para el calculo de la superficie de la poligonal en este caso un triangulo, utilizaremos la

siguiente expresin: CsenBCAC

Area = 2

; siendo BCy AC , distancias horizontales,

las cuales sern calculadas con la siguiente expresin: 22 hDidhz D-= Calculo de hD A C

mDi

mH

mhbmhb

mhimhi

Z

A

074,17893

10,3958

00,0 25,1

46,1 40,1

''00,16 '27 89 Z ''20,30 '41 90

12

21

21

==

======

Correccin de los ngulos cenitales:

( ) ( )

( ) ( )''83,16

1014,8481368117893,074''00,16 '27 89 46,100,0

''1

''73,11014,8481368117893,074

''20,30 '41 90 40,125,1''1

6-221

2

6-112

1

-=

-=

-=

-=

-=

-

=

senarcDi

senZhihb

senarcDi

senZhihb

q

q

''17,59 '26 89''83,16''00,16 '27 89

''28,47 '41 90''73,1''20,30 '41 90

2222

1111

=-==

=-==

cZZcZ

cZZcZ

q

q

( ) ( ) msencZcZsenDih 85,193''47,28 '41 90''17,59 '26 8921

074,1789321

12 -=-=-=D

Por lo tanto mHhHHmh CAC 25,3764 85,193 =\D=-=D C B

mDi

mH

mhbmhb

mhimhi

Z

C

20,14728

25,3764

34,1 34,1

45,1 46,1

''00,37 '53 89 Z ''00,51 '13 90

12

21

21

==

======



( ) ( )

( ) ( )''54,1

1014,8481368114728,20''00,37 '53 98 45,134,1

''1

''68,11014,8481368114728,20

''00,51 '13 90 46,134,1''1

6-221

2

6-112

1

-=

-=

-=

-=

-=

-

=

senarcDi

senZhihb

senarcDi

senZhihb

q

q

''35,46 '53 89''54,1''00,37 '53 89

''32,49 '13 90''68,1''00,51 '13 90

2222

1111

=-==

=-==

cZZcZ

cZZcZ

q

q

( ) ( ) msencZcZsenDih 34,43''32,49 '13 90''46,35 '53 8921

70,1472821

12 -=-=-=D

Por lo tanto mHhHHmh BCB 91,3720 34,43 =\D=-=D HB fijo = 3720,89m HB calculado = 3720,91m Error de cierre = -0,02m Tolerancia para el error de cierre de una nivelacin trigonomtrica.

.621774,32774,32621L INCLINADADISTANCIA Kmm ==S=

=

=

m

m

86,08567321139,0621774,3215,0L15,0

57,05711547426,0621774,321,0L1,0orden I

El error de cierre de altura se encuentra dentro la tolerancia por lo tanto se puede compensar.

El factor de compensacin ser:L

cierre deerror =FC

710130874428,6774,32621

02,0 --=-=FC

Compensacin de las cotas: recorrida DIn FCC S=

( ) mFCCmFCC

B

C

02,070,14728074,17893

01,080109700189,0074,17893

-=+=--==

Por lo tanto la cota corregida seria:

nncorregidan CHH =

mHH

mHH

BCBC

CCCC

89,372002,091,3720

24,376401,025,3764

=-==-=


Por lo tanto los hD son:

mhHHh

mhHHh

BCCCBCBC

CAACCCA

35,43

86,193

=D-=D=D-=D

Calculo de distancia horizontal:

mhz

mhz

CB

AC

636,1472835,4370,14728

024,1789286,193074,1789322

22

=-=D

=-=D

Calculo de la superficie:

250,131335952''30,42 '36 94 2

mAsenhzhz

A CBAC =DD

=

calculo del exceso esfrico:

''985,18 '24 312

-=+

= BAmjj

j

( )( )

( )( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

62,6384217''985,18 '24 31 220067226700.01

6378388

1

373,6352895''985,18 '24 31 220067226700.01

220067226700,016378388

1

1

2122

122

23

223

22

2

=--

=-

=

=--

-=

-

-=

j

jr

''322,03

''67,0''

''67,010848136811,4373,635289562,6384217

50,131335952''1

'' 6

+==

=

=

= -

U

mm

E

arcNArea

Er

Cierre angular de la poligonal quedando los ngulos compensados y esfricos

'0,67' 0' 180 '0,66' 0' 180 180

''52,42 '36 94: ''52,42 '36 94: 322,0 ''30,42 '36 94:

''52,24 '48 47: ''52,24 '48 47: 322,0 ''30,24 '48 47:

''63,53 '34 37: '0,01' ''62,53 '34 37: 322,0 ''40,53 '34 37:

=S=S=S++++

CCC

BBB

AAA


Calculo de azimut inverso ABa

'46,82' 4' 0 ''33,21 '12 0

'37,12 '37 70 : '65,29 '30 31 :

''55,25 '32 70 : ''32,8 '18 31 :

A

B

=D=D------

ljljlj

A

B

'286,82''' ''33,741'' =D=D lj

''41,1432

'' ''665,370

2'' =D=D lj

''985,18 '24 312

''65,29 '30 31''32,8 '18 312

-=--=+

= mBA

m jjj

j

610848136811,4''1 -=arc

220067226700,02971

2971

22 22

22 =

-=-= effe

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

62,6384217''985,81 '42 31 220067226700.01

6378388

1

373,6352895''985,81 '42 31 220067226700.01

220067226700,016378388

1

1

21

221

22

23

223

22

2

=--

=-

=

=--

-=-

-=

j

jr

( )

( ) ( )

576,993527 21

24''1''82,286

1''1''985,18 '24 31cos62,6384217''82,286

24''1''

1''1cos''21

1

2

2

1

=

D+

--=

D-D=

D+

aa

ljlaa

senS

arcarc

arcarcNsenS mm

( )

( ) ( )

7283,2283221

cos

62,638421724''1''33,741373,6352895

1''1''3,412 '2 0cos373,6352895''33,741

24

''1''1''1

21

cos''21

cos

2

2

22

2

22

2

=

D+

-=

D

-DD=

D+

aa

jrlrjaa

S

arcarc

N

arcarcS

m

mm


( )

( )

D

-DD

D-D

=

D+=

D+

D+

=

2

22

2

2

1

21

24

''1''1

21

cos''

24''1''

1cos''

21

21

cos

21

: tienese ; Si

m

mm

mm

N

arc

arcN

tgS

senS

SS

jrlrj

ljl

aaaa

aa

''94,29 '21 183318479255,021

3318479255,07283,22832

99352,757621

1 ==

D+

==

D+

-tg

tg

aa

aa

( )

( )

''73,14 '1 02

''7295,742

''''459,149''

''4594806,149''''67,10 '6 0cos

''985,18 '24 31''82,286

21

cos

''''

=D=D=D

-=D--=D

D=D-

aaa

aj

jla sen

sen m

''67,44 '22 18''73,14 '1 0''94,29 '21 18 =+= aa

Por lo tanto ''67,44 '22 198180 =+= ABAB aaa

( ) mSsensen

S 11,24057''94,29 '21 18

99352,7576

21

99352,757611 ==

D+

=aa

( ) mSS 11,24057''94,29 '21 18cos7283,22832

21

cos

7283,2283222 ==

D+

=aa

21 SS = ; Por lo tanto la distancia geodsica es: mS AB 11,24057=


Calculo de las distancias geodsicas preliminares Calculo de distancia geodsica ACS

''65,29 '30 31-== Am jj ( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

017,6384252''65,29 '30 31 220067226700.01

6378388

1

058,6352998''65,29 '30 31 220067226700.01

220067226700,016378388

1

1

2122

122

23

223

22

2

=--

=-

=

=--

-=

-

-=

j

jr

''04,51 '47 160''63,53 '34 37''67,44 '22 198 =-=-= ACABAC A aaa

( ) ( )

mR

sen

Nsen

NR

ACmACm

mm

35,6356364

''04,51 '47 160cos017,6384252''04,51 '47 160058,6352998

058,6352998017,6384252

cos

22

22

=

+=

+

=

a

aarr

a

mR

hmDhCnmm

mHH

hm

mhDiDh

CA

869,1035,6356364

17,3861024,17892

17,38612

24,376410,39582

024,1789286,193074,17893 2222

-=

-=-

=

=+=+

=

=-=D-=

a

mDnmm 155,17881869,10024,17892 =-=

mSS

mR

DnmmCc

161,17881006,0155,17881

006,035,635636424

155,11788124 2

3

2

3

=+=

=

=

=a


Calculo de distancia geodsica CBS

''65,29 '30 31-== Am jj ( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

017,6384252''65,29 '30 31 220067226700.01

6378388

1

058,6352998''65,29 '30 31 220067226700.01

220067226700,016378388

1

1

2122

122

23

223

22

2

=--

=-

=

=--

-=

-

-=

j

jr

''52,8 '11 246''52,42 '36 94180''04,51 '47 160 180 =-+=-+= CBACCB C aaa

( ) ( )

mR

sen

Nsen

NR

CBmCBm

mm

622,6379135

''52,8 '11 246cos017,6384252''52,8 '11 246058,6352998

058,6352998017,6384252

cos

22

22

=

+=

+

=

a

aarr

a

mR

hmDhCnmm

mHH

hm

mhDiDh

CB

641,8622,6379135

565,3742636,14728

565,37422

24,376489,37202

636,1472835,4370,14728 2222

-=

-=-

=

=+=+

=

=-=D-=

a

mDnmm 995,14719641,8636,14728 =-=

mSS

mR

DnmmCc

998,14719003,0995,14719

003,0622,637913524

995,1471924 2

3

2

3

=+=

=

=

=a


Calculo de posicin por el problema directo (preliminar) ( A C)

Calculo de latitud

( )444444 3444444 21444 3444 2143421

III

m

mACAC

II

m

mAC

I

m

AC

arcN

tgsenSarcN

tgsenSarc

S

''16

31cos''12''1

cos'' 2

1

223

1

22

+

-

+

=D-r

jaar

jar

aj

Iteracin 1

mSAC

161,17881

''05,51 '47 160

==a

''65,29 '30 311 -== Am jj ( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

017,6384252''65,29 '30 31 220067226700.01

6378388

1

058,6352998''65,29 '30 31 220067226700.01

220067226700,016378388

1

1

2122

1

122

1

23

223

122

2

1

=--

=-

=

=--

-=

-

-=

j

jr

( ) ''252557,5481014,8481368186352998,05''05,51 '47 160cos161,17881

''1cos

6-1

1 -==

=arc

SI

m

AC

ra

( )( )

( ) ''60539144595,0

1014,8481368186352998,0576384252,012''65,29 '30 31 ''05,51 '47 160 161,17881

''12

1

6-

22

11

122

1

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mAC

rja

( )( )

( ) ( )( )

( ) ''0330001649597,0

1014,8481368186352998,0576384252,016''65,29 '30 31 g31''05,51 '47 160cos''05,51 '47 160 161,17881

''16

31cos

1

6-2

223

12

1

1223

1

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mACAC

rjaa


( ) ( ) ( )

''31,8 '9 0

''306,548000,0''054,0''252,548'' 111

=D\

-=+--=-+=D-

j

j IIIII

''34,21 '21 31 ''31,8 '9 0''65,29 '30 31 -=\+-=D+= CCAC jjjjj

''50,55 '25 312

''34,21 '21 31''65,29 '30 3122

-=--=+

= CAmjj

j

Iteracin 2

( )( )

( )( )( )

mN

msensene

ea

m

m

017,6384252

092,6352922''50,55 '25 31 220067226700.01

220067226700,016378388

1

1

1

23

223

222

2

2

=

=--

-=-

-=j

r

( ) ''25911128,5481014,8481368126352922,09''05,51 '47 160cos161,17881

''1cos

6-2

2 -==

=arc

SI

m

AC

ra

( )( )

( ) ''30537544080,0

1014,8481368126352922,0976384252,012''50,55 '25 31 ''05,51 '47 160 161,17881

''12

2

6-

22

21

222

2

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mAC

rja

( )( )

( ) ( )( )

( ) ''5450001644413,0

1014,8481368126352922,0976384252,016''50,55 '25 31 g31''05,51 '47 160cos''05,51 '47 160 161,17881

''16

31cos

2

6-2

223

22

1

2223

2

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mACAC

rjaa

( ) ( ) ( )

''31,8 '9 0

''313,548000,0''054,0''259,548'' 222

=D\

-=+--=-+=D-

j

j IIIIII

''34,21 '21 31 ''31,8 '9 0 ''65,29 '30 31 -=\+-=D+= CCAC jjjjj


''50,55 '25 312

''34,21 '21 31''65,29 '30 3123

-=--=+

= CAmjj

j

32 mm jj = ; Por lo tanto se cumple la convergencia Calculo de longitud

( ) ( )( )m

sensene

aN

C

158,6384201''34,21 '12 31 220067226700.01

6378388

1 21

221

222 =

--=

-=

j

''51,42 '3 0''51,222'''''5128601,222

10848136811,4''34,21 '21 31cos158,6384201''05,51 '47 160 161,17881

''1cos'' 6

2

-=D\-=D=

-=

=D- -

ll

ja

l senarcN

senS

C

AC

Por lo tanto la longitud es:

''88,54 '40 70''51,42 '3 0 ''37,21 '37 70 -=\--=D+= CAC llll Calculo de azimut inverso CAa

aaa D+= 180ACCA

''51,222

'155,34 '4 02

''31,8 '9 0

=-=D

-=D-=-=D

CA

CA

lll

jjjj

( )

( )

''04,56 '1 0''04,116''

''0362871,116''''155,34 '4 0cos

''50,55 '25 31''51,222

21

cos

''''

=D=D

-=D--

-=D

D=D-

aa

aj

jla sen

sen m

''09,47 '49 340''04,56 '1 0180''05,51 '47 160180 =++=D+= CAACCA aaaa


Calculo de posicin por el problema directo ( C B)

Calculo de latitud

( )444444 3444444 21444 3444 2143421

III

m

mCBCB

II

m

mCB

I

m

CB

arcN

tgsenSarcNtgsenS

arc

S

''16

31cos''12''1

cos'' 2

1

223

1

22

+

-

+

=D-r

jaar

jar

aj

Iteracin 1

mS

CBCACB998,14719

''57,4 '13 246'42,52' 36' 94'47,09' 49' 340C

==-=-=

''34,21 '21 311 -== Cm jj ( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

158,6384201''34,21 '21 31 220067226700.01

6378388

1

228,6352846''34,21 '21 31 220067226700.01

220067226700,016378388

1

1

2122

1

122

1

23

223

122

2

1

=--

=-

=

=--

-=

-

-=

j

jr

( ) ''7296143,1921014,8481368186352846,22''57,4 '13 246cos998,14719

''1cos

6-1

1 -==

=arc

SI

m

CB

ra

( )( )

( ) ''2811411379,0

1014,8481368186352846,2286384201,152''34,21 '21 31 ''57,4 '13 246 998,14719

''12

1

6-

22

11

122

1

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mCB

rja

( )( )

( ) ( )( )

( ) ''4310003022806,0

1014,8481368186352846,2286384201,156''34,21 '21 3131''57,4 '13 246cos''57,4 '13 246 998,14719

''16

31cos

1

6-2

223

12

1

1223

1

-=

-+=

+=

III

tgsen

arcN

tgsenSIII

m

mCBCB

rjaa

( ) ( ) ( )

''01,13 '3 0

''011,193000,0''281,0''730,192'' 111

=D\

-=+--=-+=D-

j

j IIIII


''33,8 '18 31 ''01,13 '3 0''34,21 '21 31 -=\+-=D+= BBCB jjjjj Iteracin 2

''835,44 '19 312

''34,21 '21 31''33,8 '18 3122

-=--=+

= CBmjj

j

( )

( )( )

( )( )

mN

msensene

ea

m

m

158,6384201

549,6352819''835,44 '19 31 220067226700.01

220067226700,016378388

1

1

1

23

223

222

2

2

=

=--

-=-

-=j

r

( ) ''7304237,1921014,8481368196352819,54''57,4 '13 246cos998,14719

''1cos

6-2

2 -==

=arc

SI

m

CB

ra

( )( )

( ) ''2808463838,0

1014,8481368196352819,5486384201,152''835,44 '19 31''57,4 '13 246 998,14719

''12

2

6-

22

21

222

2

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mCB

rja

( )( )

( ) ( )( )

( ) ''5630003019467,0

1014,8481368196352819,5486384201,156''835,44 '19 31 g31''57,4 '13 246cos''57,4 '13 246 998,14719

''16

31cos

2

6-2

223

22

1

2223

2

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mCBCB

rjaa

( ) ( ) ( )

''01,13 '3 0

''011,193000,0''281,0''730,192'' 222

=D\

-=+--=-+=D-

j

j IIIIII

''33,8 '18 31 ''01,13 '3 0''34,21 '21 31 -=\+-=D+= BBCB jjjjj

''835,44 '19 312

''34,21 '21 31''33,8 '18 3123

-=--=+

= CBmjj

j

32 mm jj = ; Por lo tanto se cumple la convergencia.


Calculo de longitud

( ) ( )( )m

sensene

aN

B

288,6384183''33,8 '18 31 220067226700.01

6378388

1 21

221

222 =

--=

-=

j

''34,29 '8 0''34,509'''''3413639,509

10848136811,4''33,8 '18 31cos288,6384183''57,4 '13 246 998,14719

''1cos'' 6

2

=D\=D-=

-=

=D- -

ll

ja

l senarcN

senS

B

CB


''54,52 '32 70''34,29 '8 0 ''88,54 '40 70 -=\+-=D+= BCB llll

'0,01'- '' ''01,0''

''54,25 '32 70 ''33,8 '18 31

''55,25 '32 70 ''32,8 '18 31

=D=D-=-=

-=-=

ljljlj

CALCULADOBCALCULADOB

FIJOBFIJOB

Error en posicin = sS+ 22 elej

Arco de paralelo = ''1 cos arcN mm j Arco de meridiano = ''1 arcm r

77,19 '23 31

''34,21 '21 31

''33,8 '18 31

''65,29 '30 31

-=-=-=-=

m

C

B

A

jjjj

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

131,6384212''77,19 '23 31 220067226700.01

6378388

1

987,6352878''77,19 '23 31 220067226700.01

220067226700,016378388

1

1

2122

122

2322

322

2

=--

=-

=

=--

-=-

-=

j

jr

Arco de paralelo = marcarcN mm 422,26''1 ''77,19 '23 31cos131,6384212''1 cos =-= j Arco de meridiano = marcarcm 800,30''1 987,6352878''1 ==r

0,26426,422'-0,01'paralelo de Arco''

308,0800,30''01,0meridiano de Arco''

-==D===D=

leljej


Error lineal = ( ) ( ) m406,0264,0308,0 2222 =-+=+elej

mSSs CBAC 159,32601998,14719161,17881 =+=+=S

Error de posicin = 42118,80298

1

406,0159,32601

406,0406,0

159,32601406,0linealError ===

Ss

Error en posicin = 000.40122

S+s

elej; para que el error de posicin se encuentre

dentro de la tolerancia de I orden.

Error en posicin = 000025,044980000124535,0000.401

42118,802981


CALCULO DEFINITIVO Calculo del exceso esfrico

77,19 '23 31

''33,21 '21 31

''32,8 '18 31

''65,29 '30 31

-=-=-=-=

m

C

B

A

jjjj

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

131,6384212''77,19 '23 31 220067226700.01

6378388

1

987,6352878''77,19 '23 31 220067226700.01

220067226700,016378388

1

1

2122

122

2322

322

2

=--

=-

=

=--

-=-

-=

j

jr

''322,03

''67,0''

''67,010848136811,4987,6352878131,6384212

50,131335952''1

'' 6

+==

=

=

= -

U

mm

E

arcNArea

Er

Reduccin de los ngulos horizontales al elipsoide Correccin por efecto de la altura de la estacin observada.

( ) ''1 1coscos

''22

222

arceNseneH

Xm -

= aaj

''49,55 '25 312

''05,51 '47 160

24,3764; ''33,21 '21 31''65,29 '30 31

-=+=

==-=-=

CAm

AC

CCA mH

jjj

ajj

( ) ( )( )m

sensene

aN

m

m 57,6384226''49,55 '25 31 220067226700.01

6378388

1 21

221

22=

--=

-=

j


( )

( )( )

''186,0''

10848136811,4220067226700,0157,6384226''05,51 '47 160''05,51 '47 160cos''33,21 '21 31cos220067226700,024,3764

''

''1 1

coscos''

6

2

2

22

-=

--=

-

=

-

C

C

m

ACACCCC

X

senX

arceN

seneHX

aaj

''985,18 '24 312

''67,44 '22 198

89,3720; ''32,8 '18 31''65,29 '30 31

-=+

=

==-=-=

BAm

AB

BBA mH

jjj

ajj

( ) ( )( )m

sensene

aN

m

m 62,6384217''985,18 '24 31 220067226700.01

6378388

1 21

221

22=

--=

-=

j

( )

( )( )

''178,0''

10848136811,4220067226700,0162,6384217''65,29 '22 198''65,29 '22 198cos''32,8 '18 31cos220067226700,089,3720

''

''1 1

coscos''

6

2

2

22

=

--=

-

=

-

B

B

m

ABABBBB

X

senX

arceN

seneHX

aaj

ngulos observados en terreno Estacin Punto visado Angulo ledo X Angulo corregido Angulo reducido

A C 00 00 00 -0,186 359 59 59,81 00 00 00 B 37 34 53,1 1,178 37 34 54,28 37 34 54,47


''825,44 '19 312

''57,4 '13 246

89,3720; ''32,8 '18 31''33,21 '21 31

-=+=

==-=-=

BCm

CB

BBC mH

jjj

ajj

( ) ( )( )m

sensene

aN

m

m 22,6384192''825,44 '19 31 220067226700.01

6378388

1 21

221

22=

--=

-=

j

( )

( )( )

''219,0''

10848136811,4220067226700,0122,6384192''57,4 '13 246''57,4 '13 246cos''32,8 '18 31cos220067226700,089,3720

''

''1 1

coscos''

6

2

2

22

=

--=

-

=

-

B

B

m

CBCBBBB

X

senX

arceN

seneHX

aaj

''49,55 '25 312

''09,47 '49 340

10,3958; ''65,29 '30 31''33,21 '21 31

-=+=

==-=-=

ACm

CA

BAC mH

jjj

ajj

( ) ( )( )m

sensene

aN

m

m 17,6384226''49,55 '25 31 220067226700.01

6378388

1 21

221

22=

--=

-=

j

( )

( )( )

''195,0''

10848136811,4220067226700,0117,6384226''09,47 '49 340''09,47 '49 340cos''65,29 '30 31cos220067226700,010,3958

''

''1 1

coscos''

6

2

2

22

-=

--=

-

=

-

A

A

m

CACAAAA

X

senX

arceN

seneHX

aaj


C B 00 00 00 0,219 00 00 0,22 00 00 00 A 94 36 42,00 -0,195 94 36 41,80 94 36 41,58


''98,18 '24 312

''21,15 '20 18 ''46,29 '2 0180''67,44 '22 198 180

10,3958; ''65,29 '30 31''32,8 '18 31

-=+

=

=\--=D-==-=-=

ABm

BAABBA

AAB mH

jjj

aaaajj

( ) ( )( )m

sensene

aN

m

m 62,6384219''98,18 '24 31 220067226700.01

6378388

1 21

221

22=

--=

-=

j

( )

( )( )

''188,0''

10848136811,4220067226700,0162,6384219''21,15 '20 18''21,15 '20 18cos''65,29 '30 31cos220067226700,010,3958

''

''1 1

coscos''

6

2

2

22

=

--=

-

=

-

A

A

m

BABAAAA

X

senX

arceN

seneHX

aaj

''825,44 '19 312

''73,39 '8 66 ''52,24 '48 47 ''21,15 '20 18

24,3764; ''33,21 '21 31''32,8 '18 31

-=+=

=\+=+==-=-=

CBm

BCBCBABC

CCB

B

mH

jjj

aaaajj

( ) ( )( )m

sensene

aN

m

m 22,6384192''825,44 '19 31 220067226700.01

6378388

1 21

221

22=

--=

-=

j

( )

( )( )

''222,0''

10848136811,4220067226700,0122,6384192''73,39 '8 66''73,39 '8 66cos''33,21 '21 31cos220067226700,024,3764

''

''1 1

coscos''

6

2

2

22

=

--=

-

=

-

C

C

m

BCBCCCC

X

senX

arceN

seneHX

aaj


B A 00 00 00 0,188 00 00 0,19 00 00 00 C 47 48 24,00 0,222 47 48 24,22 47 48 24,03


Estacin Punto visado Angulo ledo X Angulo corregido Angulo reducido

A C 00 00 00 -0,186 359 59 59,81 00 00 00 B 37 34 53,1 1,178 37 34 54,28 37 34 54,47

B A 00 00 00 0,188 00 00 0,19 00 00 00 C 47 48 24,00 0,222 47 48 24,22 47 48 24,03

C B 00 00 00 0,219 00 00 0,22 00 00 00 A 94 36 42,00 -0,195 94 36 41,80 94 36 41,58

ngulos interiores reducidos

''08,0 '0 180 ''58,41 '36 94:

''03,24 '48 47:

''47,54 '34 37:

=S

C

B

A

''602,0 3

''08,0 ''08,0 '0 0 -=-=-= unitarioErrorError

.estaciones de numero n""con ; ''46,33''2n''2

estacin ''1orden I

=

El error de cierre angular se encuentra dentro la tolerancia por lo tanto se puede compensar.

180 '59,99' 59' 179 ''08,0 '0 180

'41,55' 36' 94:C '41,55' 36' 94:C ''602,0 '41,58' 36' 94:C

'24,00' 48' 47: '24,00' 48' 47: ''602,0 '24,03' 48' 47:

''45,54 '34 37: '0,01' ''44,54 '34 37: ''602,0 ''47,54 '34 37:

=S=S=S--+-

BBB

AAA

Cierre angular de la poligonal quedando los ngulos compensados y esfricos.

''322,03

''67,0'' +==UE

'0,67' 0' 180 '0,66' 0' 180 180

''77,41 '36 94: ''77,41 '36 94: 322,0 ''55,41 '36 94:

''22,24 '48 47: ''22,24 '48 47: 322,0 ''00,24 '48 47:

''68,54 '34 37: '0,01' ''67,54 '34 37: 322,0 ''45,54 '34 37:

=S=S=S++++

CCC

BBB

AAA


Calculo de alturas definitivas A C

mS

mH

mhbmhb

mhimhi

Z

A

161,17881

10,3958

00,0 25,1

46,1 40,1

''00,16 '27 89 Z ''20,30 '41 90

12

21

21

==

======


( ) ( )

( ) ( )''84,16

1014,8481368117881,16146,100,0

''1

''73,11014,8481368117881,161

40,125,1''1

6-21

2

6-12

1

-=

-=

-=

-=

-=

-

=

arcShihb

arcShihb

q

q

''16,59 '26 89''84,16''00,16 '27 89

''28,47 '41 90''73,1''20,30 '41 90

2222

1111

=-==

=-==

cZZcZ

cZZcZ

q

q

( ) ( ) mtgcZcZtgSh 73,193''47,28 '41 90''16,59 '26 8921

161,1788121

12 -=-=-=D

''49,55 '25 312

-=+

= CAmjj

j

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

57,6384226''49,55 '25 31 220067226700.01

6378388

1

089,6352922''49,55 '25 31 220067226700.01

220067226700,016378388

1

1

2122

122

23

223

22

2

=--

=-

=

=--

-=

-

-=

j

jr

''99,49 '47 160''68,54 '34 37''67,44 '22 198 =-=-= ACABAC A aaa


( ) ( )

mR

sen

Nsen

NR

ACmACm

mm

897,6356293

''99,49 '47 160cos57,6384226''99,49 '47 160089,6352922

089,635292257,6384226

cos

22

22

=

+=

+

=

a

aarr

a

000000659,1987,635629312

161,178811

121

000015239,1897,63562932

73,1931

21

000622706,1897,6356293

10,395811

2

2

2

2

1

=

+=

+=

=

+=

D

+=

=

+=

+=

a

a

a

RS

C

R

hB

Rh

A

mCBAhh 85,193000000659,1000015239,1000622706,173,193' -=-=D=D

mHhHH CAC 25,376485,19310,3958' =-=D=

C B

mS

mH

mhbmhb

mhimhi

Z

C

998,14719

25,3764

34,1 34,1

45,1 46,1

''00,37 '53 89 Z ''00,51 '13 90

12

21

21

==

======


( ) ( )

( ) ( )''54,1

1014,8481368114719,99845,134,1

''1

''68,11014,8481368114719,998

46,134,1''1

6-21

2

6-12

1

-=

-=

-=

-=

-=

-

=

arcShihb

arcShihb

q

q


''35,46 '53 89''54,1''00,37 '53 89

''32,49 '13 90''68,1''00,51 '13 90

2222

1111

=-==

=-==

cZZcZ

cZZcZ

q

q

( ) ( ) mtgcZcZtgSh 31,43''32,49 '13 90''46,35 '53 8921

998,1471921

12 -=-=-=D

''825,44 '19 312

-=+

= BCmjj

j

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

22,6384192''825,44 '19 31 220067226700.01

6378388

1

546,6352819''825,44 '19 31 220067226700.01

220067226700,016378388

1

1

2122

122

23

223

22

2

=--

=-

=

=--

-=

-

-=

j

jr

''32,5 '13 246''77,41 '36 94''09,47 '49 340 =-=-= CBCACB C aaa

( ) ( )

mR

sen

Nsen

NR

ACmACm

mm

472,6379069

''32,5 '13 246cos22,6384192''32,5 '13 246 546,6352819

546,635281922,6384192

cos

22

22

=

+=

+

=

a

aarr

a

000000444,1472,637906912

998,147191

121

000003395,1472,63790692

31,431

21

000590094,1472,6379069

25,376411

2

2

2

2

1

=

+=

+=

=

+=

D

+=

=

+=

+=

a

a

a

RS

C

R

hB

Rh

A

mCBAhh 34,43000000444,1000003395,1000590094,131,43' -=-=D=D

mHhHH BCB 91,372034,4325,3764' =-=D=


HB fijo = 3720,89m HB calculado = 3720,91m Error de cierre = -0,02m Tolerancia para el error de cierre de una nivelacin trigonomtrica.

.601159,32159,32601L GEODSICADISTANCIA Kmm ==S=

=

=

m

m

86,08564613696,0601159,3215,0L15,0

57,05709742464,0601159,321,0L1,0orden I

El error de cierre de altura se encuentra dentro la tolerancia por lo tanto se puede compensar.

El factor de compensacin ser:L

cierre deerror =FC

710134751222,6159,3260102,0 --=-=FC

Compensacin de las cotas: recorrida Sn FCC S=

( ) mFCCmFCC

B

C

02,0998,14719161,17881

01,0010964743,0161,17881

-=+=--==

Por lo tanto la cota corregida seria:

nncorregidan CHH =

mHH

mHH

BCBC

CCCC

89,372002,091,3720

24,376401,025,3764

=-==-=


Calculo de distancias geodsicas definitivas Calculo de distancia geodsica ACS

''49,55 '25 312

-=+

= CAmjj

j

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

57,6384226''49,55 '25 31 220067226700.01

6378388

1

089,6352922''49,55 '25 31 220067226700.01

220067226700,016378388

1

1

2122

122

23

223

22

2

=--

=-

=

=--

-=

-

-=

j

jr

''99,49 '47 160''68,54 '34 37''67,44 '22 198 =-=-= ACABAC A aaa

( ) ( )

mR

sen

Nsen

NR

ACmACm

mm

897,6356293

''99,49 '47 160cos57,6384226''99,49 '47 160089,6352922

089,635292257,6384226

cos

22

22

=

+=

+

=

a

aarr

a

mR

hmDhCnmm

mHH

hm

mhDiDh

CA

869,10987,6356293

17,3861024,17892

17,38612

24,376410,39582

024,1789286,193074,17893 2222

-=

-=-

=

=+=+

=

=-=D-=

a

mDnmm 155,17881869,10024,17892 =-=

mSS

mR

DnmmCc

161,17881006,0155,17881

006,035,635636424

155,11788124 2

3

2

3

=+=

=

=

=a


Calculo de distancia geodsica CBS

''825,44 '19 312

-=+

= BCmjj

j

( )( )

( )( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

22,6384192''825,44 '19 31 220067226700.01

6378388

1

546,6352819''825,44 '19 31 220067226700.01

220067226700,016378388

1

1

2122

122

2322

322

2

=--

=-

=

=--

-=-

-=

j

jr

''32,5 '13 246''77,41 '36 94''09,47 '49 340 =-=-= CBCACB C aaa

( ) ( )

mR

sen

Nsen

NR

ACmACm

mm

472,6379069

''32,5 '13 246cos22,6384192''32,5 '13 246 546,6352819

546,635281922,6384192

cos

22

22

=

+=

+

=

a

aarr

a

mR

hmDhCnmm

mHH

hm

mhDiDh

CB

641,8472,6379069

565,3742636,14728

565,37422

24,376489,37202

636,1472835,4370,14728 2222

-=

-=-

=

=+=+

=

=-=D-=

a

mDnmm 995,14719641,8636,14728 =-=

mSS

mR

DnmmCc

998,14719003,0995,14719

003,0622,637913524

995,1471924 2

3

2

3

=+=

=

=

=a


Calculo de posicin por el problema directo (definitivo) ( A C)

Calculo de latitud

( )444444 3444444 21444 3444 2143421

III

m

mACAC

II

m

mAC

I

m

AC

arcN

tgsenSarcN

tgsenSarc

S

''16

31cos''12''1

cos'' 2

1

223

1

22

+

-

+

=D-r

jaar

jar

aj

Iteracin 1

mSAC

161,17881

''49,49 '47 160

==a

''65,29 '30 311 -== Am jj ( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

017,6384252''65,29 '30 31 220067226700.01

6378388

1

058,6352998''65,29 '30 31 220067226700.01

220067226700,016378388

1

1

2122

1

122

1

23

223

122

2

1

=--

=-

=

=--

-=

-

-=

j

jr

( ) ''2515849,5481014,8481368186352998,05''99,49 '47 160cos161,17881

''1cos

6-1

1 -==

=arc

SI

m

AC

ra

( )( )

( ) ''0539160356,0

1014,8481368186352998,0576384252,012''65,29 '30 31 ''99,49 '47 160 161,17881

''12

1

6-

22

11

122

1

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mAC

rja

( )( )

( ) ( )( )

( ) ''330001649642,0

1014,8481368186352998,0576384252,016''65,29 '30 31 g31''99,49 '47 160cos''99,49 '47 160 161,17881

''16

31cos

1

6-2

223

12

1

1223

1

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mACAC

rjaa


( ) ( ) ( )

''31,8 '9 0

''306,548000,0''054,0''252,548'' 111

=D\

-=+--=-+=D-

j

j IIIII

''34,21 '21 31 ''31,8 '9 0''65,29 '30 31 -=\+-=D+= CCAC jjjjj

Iteracin 2

''50,55 '25 31''495,55 '25 312

''34,21 '21 31''65,29 '30 3122

--=--=+

= CAmjj

j

( )

( )( )

( )( )

mN

msensene

ea

m

m

017,6384252

092,6352922''50,55 '25 31 220067226700.01

220067226700,016378388

1

1

1

23

223

222

2

2

=

=--

-=-

-=j

r

( ) ''2581407,5481014,8481368126352922,09''99,49 '47 160cos161,17881

''1cos

6-2

2 -==

=arc

SI

m

AC

ra

( )( )

( ) ''0537559794,0

1014,8481368126352922,0976384252,012''50,55 '25 31 ''99,49 '47 160 161,17881

''12

2

6-

22

21

222

2

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mAC

rja

( )( )

( ) ( )( )

( ) ''70001644458,0

1014,8481368126352922,0976384252,016''50,55 '25 31 g31''99,49 '47 160cos''99,49 '47 160 161,17881

''16

31cos

2

6-2

223

22

1

2223

2

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mACAC

rjaa

( ) ( ) ( )

''31,8 '9 0

''312,548000,0''054,0''258,548'' 222

=D\

-=+--=-+=D-

j

j IIIIII

''34,21 '21 31 ''31,8 '9 0 ''65,29 '30 31 -=\+-=D+= CCAC jjjjj


''50,55 '25 312

''34,21 '21 31''65,29 '30 3123

-=--=+

= CAmjj

j

32 mm jj = ; Por lo tanto se cumple la convergencia Calculo de longitud

( ) ( )( )m

sensene

aN

C

158,6384201''34,21 '12 31 220067226700.01

6378388

1 21

221

222 =

--=

-=

j

''52,42 '3 0''52,222'''''5161433,222

10848136811,4''34,21 '21 31cos158,6384201''99,49 '47 160 161,17881

''1cos'' 6

2

-=D\-=D=

-=

=D- -

ll

ja

l senarcN

senS

C

AC


''89,54 '40 70''52,42 '3 0 ''37,21 '37 70 -=\--=D+= CAC llll Calculo de azimut inverso CAa

aaa D+= 180ACCA

''52,222

'155,34 '4 02

''31,8 '9 0

=D-D=D

-=D-=-=D

CA

CA

l

jjjj

( )

( )

''04,56 '1 0''04,116''

''0418095,116''''155,34 '4 0cos

''50,55 '25 31''52,222

21

cos

''''

=D=D

-=D--

-=D

D=D-

aa

aj

jla sen

sen m

''09,47 '49 340''04,56 '1 0180''05,51 '47 160180 =++=D+= CAACCA aaaa


Calculo de posicin por el problema directo ( C B)

Calculo de latitud ( )

444444 3444444 21444 3444 2143421III

m

mCBCB

II

m

mCB

I

m

CB

arcN

tgsenSarcNtgsenS

arc

S

''16

31cos''12''1

cos'' 2

1

223

1

22

+

-

+

=D-r

jaar

jar

aj

Iteracin 1

mS 998,14719

''32,5 '13 246'41,77' 36' 94'47,09' 49' 340C CBCACB=

=-=-=

''34,21 '21 311 -== Cm jj ( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

158,6384201''34,21 '21 31 220067226700.01

6378388

1

228,6352846''34,21 '21 31 220067226700.01

220067226700,016378388

1

1

2122

1

122

1

23

223

122

2

1

=--

=-

=

=--

-=

-

-=

j

jr

( ) ''7280241,1921014,8481368186352846,22''32,5 '13 246cos998,14719

''1cos

6-1

1 -==

=arc

SI

m

CB

ra

( )( )

( ) ''2811420388,0

1014,8481368186352846,2286384201,152''34,21 '21 31 ''32,5 '13 246 998,14719

''12

1

6-

22

11

122

1

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mCB

rja

( )( )

( ) ( )( )

( ) ''1770003022791,0

1014,8481368186352846,2286384201,156''34,21 '21 31 g31''32,5 '13 246cos''32,5 '13 246 998,14719

''16

31cos

1

6-2

223

12

1

1223

1

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mCBCB

rjaa

( ) ( ) ( )

''01,13 '3 0

''009,193000,0''281,0''728,192'' 111

=D\

-=+--=-+=D-

j

j IIIII


''33,8 '18 31 ''01,13 '3 0''34,21 '21 31 -=\+-=D+= BBCB jjjjj Iteracin 2

''84,44 '19 31''835,44 '19 312

''34,21 '21 31''33,8 '18 3122

--=--=+

= CBmjj

j

( )

( )( )

( )( )

mN

msensene

ea

m

m

158,6384201

551,6352819''84,44 '19 31 220067226700.01

220067226700,016378388

1

1

1

23

223

222

2

2

=

=--

-=-

-=j

r

( ) ''7288334,1921014,8481368116352819,55''32,5 '13 246cos998,14719

''1cos

6-2

2 -==

=arc

SI

m

CB

ra

( )( )

( ) ''280847299,0

1014,8481368116352819,5586384201,152''84,44 '19 31 ''32,5 '13 246 998,14719

''12

2

6-

22

21

222

2

-=

-

=

=

II

tgsenarcNtgsenS

IIm

mCB

rja

( )( )

( ) ( )( )

( ) ''4980003019452,0

1014,8481368116352819,5586384201,156''84,44 '19 31 g31''32,5 '13 246cos''32,5 '13 246 998,14719

''16

31cos

2

6-2

223

22

1

2223

2

-=

-+=

+=

III

tsen

arcN

tgsenSIII

m

mCBCB

rjaa

( ) ( ) ( )

''01,13 '3 0

''01,193000,0''281,0''729,192'' 222

=D\

-=+--=-+=D-

j

j IIIIII

''33,8 '18 31 ''01,13 '3 0''34,21 '21 31 -=\+-=D+= BBCB jjjjj

''835,44 '19 312

''34,21 '21 31''33,8 '18 3123

-=--=+

= CBmjj

j

32 mm jj = ; por lo tanto se cumple la convergencia.


Calculo de longitud

( ) ( )( )m

sensene

aN

B

288,6384183''33,8 '18 31 220067226700.01

6378388

1 21

221

222 =

--=

-=

j

''34,29 '8 0''34,509'''''341834,509

10848136811,4''33,8 '18 31cos288,6384183''32,5 '13 246 998,14719

''1cos'' 6

2

=D\=D-=

-=

=D- -

ll

ja

l senarcN

senS

B

CB


''55,52 '32 70''34,29 '8 0 ''89,54 '40 70 -=\+-=D+= BCB llll

'0,00' '' ''01,0''

''55,25 '32 70 33,8 '18 31

''55,25 '32 70 ''32,8 '18 31

=D=D-=-=-=-=

ljljlj

CALCULADOBCALCULADOB

FIJOBFIJOB

Error en posicin = sS+ 22 elej

Arco de paralelo = ''1 cos arcN mm j Arco de meridiano = ''1 arcm r

77,19 '23 31

''34,21 '21 31

''33,8 '18 31

''65,29 '30 31

-=-=-=-=

m

C

B

A

jjjj

( )

( )( )

( )( )

( ) ( )( )m

sensene

aN

msensene

ea

m

m

m

m

131,6384212''77,19 '23 31 220067226700.01

6378388

1

987,6352878''77,19 '23 31 220067226700.01

220067226700,016378388

1

1

2122

122

2322

322

2

=--

=-

=

=--

-=-

-=

j

jr

Arco de paralelo = marcarcN mm 422,26''1 ''77,19 '23 31cos131,6384212''1 cos =-= j Arco de meridiano = marcarcm 800,30''1 987,6352878''1 ==r

00,026,422'0,00'paralelo de Arco''

308,0800,30''01,0meridiano de Arco''

==D===D=

leljej


Error lineal = ( ) ( ) m308,00308,0 2222 =+=+elej

mSSs CBAC 159,32601998,14719161,17881 =+=+=S

Error de posicin = 9188,105847

1

308,0159,32601

308,0308,0

159,32601308,0linealError ===

Ss

Error en posicin = 000.40122

S+s

elej; para que el error de posicin se encuentre

dentro de la tolerancia de I orden.

Error en posicin = 000025,0168840000094475,0000.401

9188,1058471

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