topografia 2
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UNIVERSIDAD NACIONAL DE SAN ANTONIO ABAD DEL CUSCO
TRABAJO DE TOPOGRAFÍA II 1. Ajuste de un polígono Método Topográfico
2. Compensación de Triángulos
Ingeniería Civil
Topografía II
MGT. ESCOBAR MASIAS, JUAN PABLO
1
TRABAJO DE TOPOGRAFÍA II
. Condiciones Geométricas
I1 + D1 + V1 = 200 g I2 + D2 + V2 = 200 g I3 +D3 +V3 = 200 g
200.0090 g ≠ 200 g 200.0019 g ≠ 200 g 199.9933 g ≠ 200 g
Error = 90
3 = 30 Error=
19
3 = 6.3 Error=
67
3 = 22.3
I4 + D4 + V4 = 200 g I5 + D5 +V5 = 200
200.0055 g ≠ 200 g 199.9943 g ≠ 200 g
Error= 55
3 = 18.3 Error=
57
3 = 19
Ángulos corregidos por triángulos
I1=90.5200 g - 30 D1=41.0010 g - 30 V1=68.4880 g - 30
I2=104.7269 g - 6 D2=53.3590 g - 6 V2=41.9160 g - 7 I3=63.5020 g + 22 D3=84.3718 g + 22 V3=52.1195 g + 23 I4=86.2500 g - 18 D4=60.5590 g - 18 V4=53.1965 g - 19 I5=54.9900 g + 19 D5=59.6453 g + 19 V5=85.3590 + 19
Los ángulos de I en grados centesimales (g)
Los ángulos de D en grados centesimales (g)
Los ángulos de V en grados centesimales (g)
I1=90.5200 g D1=41.0010 g V1=68.4880 g
I2=104.7269 g D2=53.3590 g V2=41.9160 g
I3=63.5020 g D3=84.3718 g V3=52.1195 g
I4=86.2500 g D4=60.5590 g V4=53.1965 g
I5=54.9900 g D5=59.6453 g V5=85.3590 g
2
TRABAJO DE TOPOGRAFÍA II
I1=90.5170 g D1=40.9980 g V1=68.4850 g
I2=104.7263 g D2=53.3584 g V2=41.9153 g I3=63.5042 g D3=84.3740 g V3=52.1218 g I4=86.2482 g D4=60.5572 g V4=53.1946 g I5=54.9919 g D5=59.64 72 g V5=85.3609 g
I1 + I2 + I3 + I4 + I5 = 400 g
399.9876 g ≠ 400 g
Error total = 0.0124
Error= 124
3 = 24.8 25
Ángulos corregidos geométricamente
I1=90.5170 g + 25 D1=40.9980 g - 12 V1=68.4850 g - 13
I2=104.7263 g + 25 D2=53.3584 g - 12 V2=41.9153 g - 13 I3=63.5042 g + 25 D3=84.3740 g - 12 V3=52.1218 g - 13 I4=86.2482 g + 25 D4=60.5572 g - 12 V4=53.1946 g - 13 I5=54.9919 g + 24 D5=59.64 72 g - 12 V5=85.3609 g - 12
I1=90.5195 g D1=40.9968 g V1=68.4837 g
I2=104.7288 g D2=53.3572 g V2=41.9140 g I3=63.5067 g D3=84.3728 g V3=52.1205 g I4=86.2507 g D4=60.5560 g V4=53.1933 g I5=54.9943 g D5=59.64 60 g V5=85.3597 g
. Condición Trigonométrica
3
TRABAJO DE TOPOGRAFÍA II
𝐴𝐹
𝑠𝑖𝑛𝑉1=
𝐵𝐹
𝑠𝑖𝑛𝐷1 ;
𝐵𝐹
𝑠𝑖𝑛𝑉2=
𝐶𝐹
𝑠𝑖𝑛𝐷2 ;
𝐶𝐹
𝑠𝑖𝑛𝑉3=
𝐷𝐹
𝑠𝑖𝑛𝐷3 ;
𝐷𝐹
𝑠𝑖𝑛𝑉4=
𝐸𝐹
𝑠𝑖𝑛𝐷4 ;
𝐸𝐹
𝑠𝑖𝑛𝑉5=
𝐴𝐹
𝑠𝑖𝑛𝐷5
Multiplicando miembro a miembro y luego simplificando
1 =𝑠𝑖𝑛𝑉1𝑠𝑖𝑛𝑉2𝑠𝑖𝑛𝑉3𝑠𝑖𝑛𝑉4𝑠𝑖𝑛𝑉5
𝑠𝑖𝑛𝐷1𝑠𝑖𝑛𝐷2𝑠𝑖𝑛𝐷3𝑠𝑖𝑛𝐷4𝑠𝑖𝑛𝐷5= ∆ Ecuación de la condición trigonométrica
∆ = 0.283919446
0.2839940229= 0.9997373995
∑ cot D = 3.930618999 ∑ cot N = 3.906704474
e = ∆−1
∑cot D + ∆ × ∑ cot N × 636620 =− 21
Ángulos finales
I1=90.5195 g D1=40.9968 g - 21 V1=68.4837 g + 21
I2=104.7288 g D2=53.3572 g - 21 V2=41.9140 g + 21
I3=63.5067 g D3=84.3728 g - 21 V3=52.1205 g + 21
I4=86.2507 g D4=60.5560 g - 21 V4=53.1933 g + 21
I5=54.9943 g D5=59.64 60 g - 21 V5=85.3597 g + 21
I1=90.5195 g D1=40.9947 g V1=68.4858 g
I2=104.7288 g D2=53.3551 g V2=41.9161 g
I3=63.5067 g D3=84.3707 g V3=52.1226 g
I4=86.2507 g D4=60.5539 g V4=53.1954 g
I5=54.9943 g D5=59.64 39 g V5=85.3618 g
Ya teniendo los ángulos corregidos geométrica y trigonométricamente calcularemos un lado por dos caminos y los
valores tienen que ser iguales
4
TRABAJO DE TOPOGRAFÍA II
𝐴𝐹
𝑠𝑖𝑛𝑉1=
𝐵𝐹
𝑠𝑖𝑛𝐷1 ;
𝐵𝐹
𝑠𝑖𝑛𝑉2=
𝐶𝐹
𝑠𝑖𝑛𝐷2
1800
𝑠𝑖𝑛68.4858=
𝐵𝐹
𝑠𝑖𝑛40.9947 ⇒ 𝐵𝐹 =
1800𝑠𝑖𝑛40.9947
𝑠𝑖𝑛68.4858 ⇒ BF = 1228.055605
1228.055605
𝑠𝑖𝑛41.9161=
𝐶𝐹
𝑠𝑖𝑛53.3551 ⇒ CF =
1228.055605𝑠𝑖𝑛53.3551
𝑠𝑖𝑛41.9161 ⇒ CF = 1492.002005
𝐴𝐹
𝑠𝑖𝑛𝐷5=
𝐸𝐹
𝑠𝑖𝑛𝑉5 ;
𝐸𝐹
𝑠𝑖𝑛𝐷4=
𝐷𝐹
𝑠𝑖𝑛𝑉4 ;
𝐷𝐹
𝑠𝑖𝑛𝐷3=
𝐶𝐹
𝑠𝑖𝑛𝑉3
1800
𝑠𝑖𝑛59.64 39 =
𝐸𝐹
𝑠𝑖𝑛85.3618 ⇒ EF =
1800𝑠𝑖𝑛85.3618
𝑠𝑖𝑛59.64 39 = EF = 2175.238572
2175.238572
𝑠𝑖𝑛60.5539=
𝐷𝐹
𝑠𝑖𝑛53.1954 ⇒ DF =
2175.238572𝑠𝑖𝑛53.1954
𝑠𝑖𝑛60.5539 = ⇒ DF = 1981.770138
1981.770138
𝑠𝑖𝑛84.3707=
𝐶𝐹
𝑠𝑖𝑛52.1226 ⇒ C F =
1981.770138𝑠𝑖𝑛52.1226
𝑠𝑖𝑛84.3707 ⇒ CF = 1491.995877
⇒CF = 1492.002005 ≈ 1491.995877
Calculando lados del polígono
𝐴𝐹
𝑠𝑖𝑛𝑉1=
𝐴𝐵
𝑠𝑖𝑛𝐼1 ⇒
1800
𝑠𝑖𝑛68.4858=
𝐴𝐵
𝑠𝑖𝑛90.5195 ⇒ AB =
1800𝑠𝑖𝑛90.5195
𝑠𝑖𝑛68.4858 ⇒ AB = 2022.913461
𝐵𝐹
𝑠𝑖𝑛𝑉2=
𝐵𝐶
𝑠𝑖𝑛𝐼2 ⇒
1228.055605
𝑠𝑖𝑛41.9161=
𝐵𝐶
𝑠𝑖𝑛104.7288 ⇒BC =
1228.055605𝑠𝑖𝑛104.7288
𝑠𝑖𝑛41.9161 ⇒ BC = 2001.534511
𝐶𝐹
𝑠𝑖𝑛𝑉3=
𝐶𝐷
𝑠𝑖𝑛𝐼3 ⇒
1492.002005
𝑠𝑖𝑛52.1226 =
𝐶𝐷
𝑠𝑖𝑛63.5067⇒ CD =
1492.002005𝑠𝑖𝑛63.5067
𝑠𝑖𝑛52.1226 ⇒ CD= 1716.460631
𝐶𝐷
𝑠𝑖𝑛𝐼3=
𝐷𝐹
𝑠𝑖𝑛𝐷3⇒
1716.460631
𝑠𝑖𝑛63.5067=
𝐷𝐹
𝑠𝑖𝑛84.3707 ⇒ DF =
1716.460631𝑠𝑖𝑛84.3707
𝑠𝑖𝑛63.5067 ⇒ DF = 1981.778277
𝐷𝐹
𝑠𝑖𝑛𝑉4=
𝐷𝐸
𝑠𝑖𝑛𝐼4 ⇒
1981.778277
𝑠𝑖𝑛53.1954=
𝐷𝐸
𝑠𝑖𝑛86.2507 ⇒ DE =
1981.778277𝑠𝑖𝑛86.2507
𝑠𝑖𝑛53.1954 ⇒ DE =2609.890167
𝐷𝐹
𝑠𝑖𝑛𝑉4=
𝐸𝐹
𝑠𝑖𝑛𝐷4 ⇒
1981.778277
𝑠𝑖𝑛53.1954=
𝐸𝐹
𝑠𝑖𝑛60.5539 ⇒ EF =
1981.778277𝑠𝑖𝑛60.5539
𝑠𝑖𝑛53.1954 ⇒ EF =2175.247506
𝐸𝐹
𝑠𝑖𝑛𝑉5=
𝐴𝐸
𝑠𝑖𝑛𝐼5 ⇒
2175.247506
𝑠𝑖𝑛85.3618=
𝐴𝐸
𝑠𝑖𝑛54.9943 ⇒ AE =
2175.247506𝑠𝑖𝑛54.9943
𝑠𝑖𝑛85.3618 ⇒ AE =1698.651646
𝐸𝐹
𝑠𝑖𝑛𝑉5=
𝐴𝐹
𝑠𝑖𝑛𝐷5 ⇒
2175.247506
𝑠𝑖𝑛85.3618=
𝐴𝐹
𝑠𝑖𝑛59.64 39 ⇒ AF =
2175.247506𝑠𝑖𝑛59.64 39
𝑠𝑖𝑛85.3618 ⇒ AF =1800.007393
⇒AF=1800 ≈1800.007393
Lados del polígono
AB=2022.913461
BC=2001.534511
CD=1716.460631
5
TRABAJO DE TOPOGRAFÍA II
DE=2609.890167
EA=1698.651646
Calculando coordenadas
VÉRTICES DISTANCIA ÁNGULOS AZIMUT A 126.3565 g B 2022.913461 30
B 121.8409 g C 2001.534511 108.1591
C 126.2868 g D 1716.460631 181.8723
D 112.6765 g E 2609.890167 269.1958
E 112.8393 g A 1698.651646 356.3565
Vértices D x cos(Azimut) D x sen(Azimut) Coordenadas +Y -Y +X -X Y X
A 52000 18000 B 1802.429092 918.3834931 53802.4291 18918.3835
B C 255.820667 1985.118683 53546.6084 20903.5022
C D 1647.34243 482.1826334 51899.266 21385.6848
D E 1214.14565 2310.2764 50685.1204 19075.4084
E A 1314.876378 1075.415 51999.9968 17999.9934
6
TRABAJO DE TOPOGRAFÍA II
α1 = 45°25ˡ14ˡˡ α2 = 44°10ˡ59ˡˡ α3 = 50°14ˡ35ˡˡ
α4 = 40°09ˡ12ˡˡ α5 = 40°55ˡ59ˡˡ α6 = 48°40ˡ22ˡˡ
α7 = 43°18ˡ47ˡˡ α8 = 47°05ˡ08ˡˡ
I1 = 57°53ˡ15ˡˡ D1 = 71°41ˡ03ˡˡ V1 =50°26ˡ08ˡˡ
I2 = 40°30ˡ34ˡˡ D2 = 76°13ˡ15ˡˡ V2 =63°16ˡ07ˡˡ
I3 = 46°26ˡ15ˡˡ D3 = 60°04ˡ36ˡˡ V3 =73°29ˡ14ˡˡ
I4 = 78°20ˡ34ˡˡ D4 = 34°50ˡ00ˡˡ V4 =66°49ˡ31ˡˡ
I5 = 80°56ˡ42ˡˡ D5 = 57°06ˡ04ˡˡ V5 =41°57ˡ17ˡˡ
I6 = 55°53ˡ15ˡˡ D6 = 65°10ˡ23ˡˡ V6 =58°56ˡ55ˡˡ
β1 = 83°23ˡ18ˡˡ β2 = 70°31ˡ20ˡˡ β3 = 26°05ˡ31ˡˡ
β4 = 29°57ˡ43ˡˡ β5 = 68°49ˡ37ˡˡ β6= 81°12ˡ35ˡˡ
. Corrección del cuadrilátero
α1 = 45°25ˡ14ˡˡ α2 = 44°10ˡ59ˡˡ α3 = 50°14ˡ35ˡˡ
α4 = 40°09ˡ12ˡˡ α5 = 40°55ˡ59ˡˡ α6 = 48°40ˡ22ˡˡ
α7 = 43°18ˡ47ˡˡ α8 = 47°05ˡ08ˡˡ
Condición Geométrica
α1 + α2 = α5 + α6 α3 + α4 = α7 + α8
89°36ˡ13ˡˡ ≠ 89°36ˡ21ˡˡ 90°23ˡ47ˡˡ ≠ 90°23ˡ55ˡˡ
Error = 8ˡˡ Error = 8ˡˡ
8
4 = 2ˡˡ
8
4= 2ˡˡ
Ángulos corregidos
α1 = 45°25ˡ14ˡˡ + 2ˡˡ α2 = 44°10ˡ59ˡˡ + 2ˡˡ α3 = 50°14ˡ35ˡˡ + 2ˡˡ
α4 = 40°09ˡ12ˡˡ + 2ˡˡ α5 = 40°55ˡ59ˡˡ - 2ˡˡ α6 = 48°40ˡ22ˡˡ - 2ˡˡ
α7 = 43°18ˡ47ˡˡ - 2ˡˡ α8 = 47°05ˡ08ˡˡ - 2ˡˡ
α1 = 45°25ˡ16ˡˡ α2 = 44°11ˡ01ˡˡ α3 = 50°14ˡ37ˡˡ
α4 = 40°09ˡ14ˡˡ α5 = 40°55ˡ57ˡˡ α6 = 48°40ˡ20ˡˡ
α7 = 43°18ˡ45ˡˡ α8 = 47°05ˡ06ˡˡ
α1 + α2 + α3 + α4 + α5 + α6 + α7 + α8 = 360°
7
TRABAJO DE TOPOGRAFÍA II
36°00016 ≠ 360°
Error = 16ˡˡ
16
8= 2ˡˡ
α1 = 45°25ˡ16ˡˡ -2ˡˡ α2 = 44°11ˡ01ˡ -2ˡˡ α3 = 50°14ˡ37ˡˡ-2ˡˡ
α4 = 40°09ˡ14ˡˡ -2ˡˡ α5 = 40°55ˡ57ˡˡ -2ˡˡ α6 = 48°40ˡ20ˡˡ-2ˡˡ
α7 = 43°18ˡ45ˡˡ-2ˡˡ α8 = 47°05ˡ06ˡˡ -2ˡˡ
Ángulos corregidos
α1 = 45°25ˡ14ˡˡ α2 = 44°10ˡ59ˡˡ α3 = 50°14ˡ35ˡˡ
α4 = 40°09ˡ12ˡˡ α5 = 40°55ˡ55ˡˡ α6 = 48°40ˡ18ˡˡ
α7 = 43°18ˡ43ˡˡ α8 = 47°05ˡ04ˡˡ
Condiciones Trigonométrica
∆ = 𝑠𝑖𝑛𝛼1𝑠𝑖𝑛𝛼3𝑠𝑖𝑛𝛼5𝑠𝑖𝑛𝛼7
𝑠𝑖𝑛𝛼2𝑠𝑖𝑛𝛼4𝑠𝑖𝑛𝛼6𝑠𝑒𝑛8 =
0.246091536
0.2471608792
∆ = 0.9956734028
∑cotD = 4.023389736 ∑cotN = 4.031185769
eˡˡ = ∆−1
∑cot D + ∆ × ∑ cot N ×206265
eˡˡ = 0.9956734028−1
4.023389736 + 0.9956734028 × 4.031185769 ×206265 = -111.037
α1 = 45°25ˡ14ˡˡ -1ˡ51ˡˡ α2 = 44°10ˡ59ˡˡ - 1ˡ51ˡˡ α3 = 50°14ˡ35ˡˡ +1ˡ51ˡˡ
α4 = 40°09ˡ12ˡˡ -1ˡ51ˡˡ α5 = 40°55ˡ55ˡˡ +1ˡ51ˡˡ α6 = 48°40ˡ18ˡˡ - 1ˡ51ˡˡ
α7 = 43°18ˡ43ˡ + 1ˡ51ˡˡ α8 = 47°05ˡ04ˡˡ -1ˡ51ˡˡ
α1 = 45°27ˡ05ˡˡ α2 = 44°09ˡ08ˡˡ α3 = 50°16ˡ26ˡˡ
α4 = 40°07ˡ21ˡˡ α5 = 40°57ˡ46ˡˡ α6 = 48°38ˡ27ˡˡ
α7 = 43°20ˡ34ˡˡ α8 = 47°03ˡ13ˡˡ
Calculo de distancias
8
TRABAJO DE TOPOGRAFÍA II
𝐴𝐵
𝑠𝑖𝑛𝛼5=
𝐵𝐶
𝑠𝑖𝑛𝛼8 ⇒ AB=1329.890m
𝐵𝐶
𝑠𝑖𝑛𝛼3=
𝐶𝐷
𝑠𝑖𝑛𝛼6 ⇒ CD= 1307.615m
𝐴𝐵
𝑠𝑖𝑛𝛼2=
𝐴𝐷
𝑠𝑖𝑛𝛼7 ⇒ AD= 1182.419m
𝐴𝐷
𝑠𝑖𝑛𝛼4=
𝐶𝐷
𝑠𝑖𝑛𝛼1⇒ CD=1307.613m
. Corrección del polígono
I1 = 57°53ˡ15ˡˡ D1 = 71°41ˡ03ˡˡ V1 =50°26ˡ08ˡˡ
I2 = 40°30ˡ34ˡˡ D2 = 76°13ˡ15ˡˡ V2 =63°16ˡ07ˡˡ
I3 = 46°26ˡ15ˡˡ D3 = 60°04ˡ36ˡˡ V3 =73°29ˡ14ˡˡ
I4 = 78°20ˡ34ˡˡ D4 = 34°50ˡ00ˡˡ V4 =66°49ˡ31ˡˡ
I5 = 80°56ˡ42ˡˡ D5 = 57°06ˡ04ˡˡ V5 =41°57ˡ17ˡˡ
I6 = 55°53ˡ15ˡˡ D6 = 65°10ˡ23ˡˡ V6 =58°56ˡ55ˡˡ
Condiciones geométricas
I1 + D1 + V1 = 180° I2 + D2 +V2 = 180° I3 + D3 +V3 = 180°
180°00ˡ26ˡˡ ≠ 180° 179°59ˡ56ˡˡ ≠ 180° 180°00ˡ05ˡˡ ≠ 180°
Error=26
3= 8.6 Error=
4
3= 1.3 Error=
5
3= 1.6
I4 + D4 + V4 = 180° I5 + D5 +V5 = 180° I6 + D6 +V6 = 180°
180°00ˡ05ˡˡ ≠ 180° 180°00ˡ03ˡˡ ≠ 180° 180°00ˡ33ˡˡ ≠ 180°
Error=5
3= 1.6 Error=
3
3= 1 Error=
33
3= 11
Ángulos corregidos por triángulos
I1 = 57°53ˡ15ˡˡ - 8ˡˡ D1 = 71°41ˡ03ˡˡ - 9ˡˡ V1 =50°26ˡ08ˡˡ - 9ˡˡ
I2 = 40°30ˡ34ˡˡ + 2ˡˡ D2 = 76°13ˡ15ˡˡ + 1ˡˡ V2 =63°16ˡ07ˡˡ + 1ˡˡ
I3 = 46°26ˡ15ˡˡ - 1ˡˡ D3 = 60°04ˡ36ˡˡ - 2ˡˡ V3 =73°29ˡ14ˡˡ - 2ˡˡ
I4 = 78°20ˡ34ˡˡ - 1ˡˡ D4 = 34°50ˡ00ˡˡ - 2ˡˡ V4 =66°49ˡ31ˡˡ -2ˡˡ
I5 = 80°56ˡ42ˡˡ - 1ˡˡ D5 = 57°06ˡ04ˡˡ -1ˡˡ V5 =41°57ˡ17ˡˡ - 1ˡˡ
I6 = 55°53ˡ15ˡˡ -11ˡˡ D6 = 65°10ˡ23ˡˡ -11ˡˡ V6 =58°56ˡ55ˡˡ -11ˡˡ
I1 = 57°53ˡ07ˡˡ D1 = 71°40ˡ54ˡˡ V1 =50°25ˡ59ˡˡ
I2 = 40°30ˡ36ˡˡ D2 = 76°13ˡ16ˡˡ V2 =63°16ˡ08ˡˡ
I3 = 46°26ˡ14ˡˡ D3 = 60°04ˡ34ˡˡ V3 =73°29ˡ12ˡˡ
I4 = 78°20ˡ33ˡˡ D4 = 34°49ˡ58ˡˡ V4 =66°49ˡ29ˡˡ
9
TRABAJO DE TOPOGRAFÍA II
I5 = 80°56ˡ41ˡˡ D5 = 57°06ˡ03ˡˡ V5 =41°57ˡ16ˡˡ
I6 = 55°53ˡ04ˡˡ D6 = 65°10ˡ12ˡˡ V6 =58°56ˡ44ˡˡ
I1 + I2 + I3 + I4 + I5 + I6 = 360°
360°00ˡ00ˡˡ ≠ 360°
Error = 15ˡˡ
15
6= 2.5
Ángulos corregidos por vértice
I1 = 57°53ˡ07ˡˡ -2ˡˡ D1 = 71°40ˡ54ˡˡ +1ˡˡ V1 =50°25ˡ59ˡˡ +1ˡˡ
I2 = 40°30ˡ36ˡˡ -3ˡˡ D2 = 76°13ˡ16ˡˡ +2ˡˡ V2 =63°16ˡ08ˡˡ +1ˡˡ
I3 = 46°26ˡ14ˡˡ -2ˡˡ D3 = 60°04ˡ34ˡˡ +1ˡˡ V3 =73°29ˡ12ˡˡ +1ˡˡ
I4 = 78°20ˡ33ˡˡ - 3ˡˡ D4 = 34°49ˡ58ˡˡ +1ˡˡ V4 =66°49ˡ29ˡˡ +2ˡˡ
I5 = 80°56ˡ41ˡˡ -2ˡˡ D5 = 57°06ˡ03ˡˡ +1ˡˡ V5 =41°57ˡ16ˡˡ +1ˡˡ
I6 = 55°53ˡ04ˡˡ -3ˡˡ D6 = 65°10ˡ12ˡˡ +2ˡˡ V6 =58°56ˡ44ˡˡ +1ˡˡ
I1 = 57°53ˡ05ˡˡ D1 = 71°40ˡ55ˡˡ V1 =50°26ˡ00ˡˡ
I2 = 40°30ˡ33ˡˡ D2 = 76°13ˡ18ˡˡ V2 =63°16ˡ09ˡˡ
I3 = 46°26ˡ12ˡˡ D3 = 60°04ˡ35ˡˡ V3 =73°29ˡ13ˡˡ
I4 = 78°20ˡ30ˡˡ D4 = 34°49ˡ59ˡˡ V4 =66°49ˡ31ˡˡ
I5 = 80°56ˡ39ˡˡ D5 = 57°06ˡ04ˡˡ V5 =41°57ˡ17ˡˡ
I6 = 55°53ˡ01ˡˡ D6 = 65°10ˡ14ˡˡ V6 =58°56ˡ45ˡˡ
Condiciones trigonométricas
∆=𝑠𝑖𝑛𝑉1𝑠𝑖𝑛𝑉2𝑠𝑖𝑛𝑉3𝑠𝑖𝑛𝑉4𝑠𝑖𝑛𝑉5𝑠𝑖𝑛𝑉6
𝑠𝑖𝑛𝐷1𝑠𝑖𝑛𝐷2𝑠𝑖𝑛𝐷3𝑠𝑖𝑛𝐷4𝑠𝑖𝑛𝐷5𝑠𝑖𝑛𝐷6=
0.3475526911
0.3478103017
∆= 0.9992593361
∑cotD = 3.698496404 ∑cotN = 3.768981433
eˡˡ = ∆−1
∑cot D + ∆ × ∑ cot N ×206265
eˡˡ = 0.9992593361−1
3.698496404 + 0.9992593361 × 3.768981433 ×206265 = -20.466 ≈ 20
Ángulos finales corregidos
I1 = 57°53ˡ05ˡˡ D1 = 71°40ˡ55ˡˡ -20ˡˡ V1 =50°26ˡ00ˡˡ +20ˡˡ
I2 = 40°30ˡ33ˡˡ D2 = 76°13ˡ18ˡˡ -20ˡˡ V2 =63°16ˡ09ˡˡ +20ˡˡ
10
TRABAJO DE TOPOGRAFÍA II
I3 = 46°26ˡ12ˡˡ D3 = 60°04ˡ35ˡˡ-20ˡˡ V3 =73°29ˡ13ˡˡ +20ˡˡ
I4 = 78°20ˡ30ˡˡ D4 = 34°49ˡ59ˡˡ -20ˡˡ V4 =66°49ˡ31ˡˡ +20ˡˡ
I5 = 80°56ˡ39ˡˡ D5 = 57°06ˡ04ˡˡ-20ˡˡ V5 =41°57ˡ17ˡˡ +20ˡˡ
I6 = 55°53ˡ01ˡˡ D6 = 65°10ˡ14ˡˡ-20ˡˡ V6 =58°56ˡ45ˡˡ+20ˡˡ
I1 = 57°53ˡ05ˡˡ D1 = 71°40ˡ35ˡˡ V1 =50°26ˡ20ˡˡ
I2 = 40°30ˡ33ˡˡ D2 = 76°12ˡ58ˡˡ V2 =63°16ˡ29ˡˡ
I3 = 46°26ˡ12ˡˡ D3 = 60°04ˡ15ˡˡ V3 =73°29ˡ33ˡˡ
I4 = 78°20ˡ30ˡˡ D4 = 34°49ˡ39ˡˡ V4 =66°49ˡ51ˡˡ
I5 = 80°56ˡ39ˡˡ D5 = 57°05ˡ44ˡˡ V5 =41°57ˡ37ˡˡ
I6 = 55°53ˡ01ˡˡ D6 = 65°09ˡ54ˡˡ V6 =58°57ˡ05ˡˡ
Calculo de distancias CD = 1307.614
𝐶𝐷
𝑠𝑖𝑛𝐼1=
𝐶𝐸
𝑠𝑖𝑛𝑉1 ⇒ CE=1190.228m
𝐼𝐸
𝑠𝑖𝑛𝐷3=
𝐸𝐻
𝑠𝑖𝑛𝑉3 ⇒ EH= 1210.978m
𝐸𝐺
𝑠𝑖𝑛𝐷5=
𝐸𝐹
𝑠𝑖𝑛𝑉5 ⇒ EF= 1552.434m
𝐷𝐸
𝑠𝑖𝑛𝐷1=
𝐶𝐸
𝑠𝑖𝑛𝑉1⇒ 𝐶𝐸 = 1190.138m
𝐸𝐼
𝑠𝑖𝑛𝑉2=
𝐶𝐼
𝑠𝑖𝑛𝐼2 ⇒ CI = 796.061m
𝐸𝐺
𝑠𝑖𝑛𝑉4=
𝐻𝐺
𝑠𝑖𝑛𝐼4 ⇒ HG= 2076.658m
𝐸𝐷
𝑠𝑖𝑛𝑉6=
𝐷𝐹
𝑠𝑖𝑛𝐼6 ⇒ DF= 1416.232m
Corrección de los triángulos
β1 = 83°23ˡ18ˡˡ β2 = 70°31ˡ20ˡˡ β3 = 26°05ˡ31ˡˡ
β4 = 29°57ˡ43ˡˡ β5 = 68°49ˡ37ˡˡ β6= 81°12ˡ35ˡˡ
Condición geométrica
Β1 + β2 + β3 = 180° Β1 + β2 + β3 = 180°
180°00ˡ09ˡˡ ≠ 180° 179°59ˡ55ˡˡ ≠ 180°
Error = 9ˡˡ Error = 5ˡˡ
9
3= 3
5
3= 1.76 ≈2
β1 = 83°23ˡ18ˡˡ - 3ˡˡ β2 = 70°31ˡ20ˡˡ - 3ˡˡ β3 = 26°05ˡ31ˡˡ - 3ˡˡ
11
TRABAJO DE TOPOGRAFÍA II
β4 = 29°57ˡ43ˡˡ + 2ˡˡ β5 = 68°49ˡ37ˡˡ + 2ˡˡ β6= 81°12ˡ35ˡˡ + 1ˡˡ
β1 = 83°23ˡ15ˡˡ β2 = 70°31ˡ17ˡˡ β3 = 26°05ˡ28ˡˡ
β4 = 29°57ˡ45ˡˡ β5 = 68°49ˡ39ˡˡ β6= 81°12ˡ36ˡˡ
Calculo de lados HG = 2076.658
𝐻𝐺
𝑠𝑖𝑛𝛽3=
𝐻𝐽
𝑠𝑖𝑛𝛽1 ⇒ HJ=4690.414m
𝐻𝐽
𝑠𝑖𝑛𝛽1=
𝐺𝐽
𝑠𝑖𝑛𝛽2 ⇒ GJ= 4451.576m
𝐺𝐽
𝑠𝑖𝑛𝛽6=
𝐺𝐾
𝑠𝑖𝑛𝛽5 ⇒ GK= 4200.416m
𝐺𝐾
𝑠𝑖𝑛𝛽5=
𝐽𝐾
𝑠𝑖𝑛𝛽4⇒ 𝐽𝐾 = 2249.649m
-------------------------------------------------------------------------------------------------------------------------------------------------------
α1 = 45°27ˡ05ˡˡ α2 = 44°09ˡ08ˡˡ α3 = 50°16ˡ26ˡˡ
α4 = 40°07ˡ21ˡˡ α5 = 40°57ˡ46ˡˡ α6 = 48°38ˡ27ˡˡ
α7 = 43°20ˡ34ˡˡ α8 = 47°03ˡ13ˡˡ
I1 = 57°53ˡ05ˡˡ D1 = 71°40ˡ35ˡˡ V1 =50°26ˡ20ˡˡ
I2 = 40°30ˡ33ˡˡ D2 = 76°12ˡ58ˡˡ V2 =63°16ˡ29ˡˡ
I3 = 46°26ˡ12ˡˡ D3 = 60°04ˡ15ˡˡ V3 =73°29ˡ33ˡˡ
I4 = 78°20ˡ30ˡˡ D4 = 34°49ˡ39ˡˡ V4 =66°49ˡ51ˡˡ
I5 = 80°56ˡ39ˡˡ D5 = 57°05ˡ44ˡˡ V5 =41°57ˡ37ˡˡ
I6 = 55°53ˡ01ˡˡ D6 = 65°09ˡ54ˡˡ V6 =58°57ˡ05ˡˡ
12
TRABAJO DE TOPOGRAFÍA II
β1 = 83°23ˡ15ˡˡ β2 = 70°31ˡ17ˡˡ β3 = 26°05ˡ28ˡˡ
β4 = 29°57ˡ45ˡˡ β5 = 68°49ˡ39ˡˡ β6= 81°12ˡ36ˡˡ
LADOS
AB=1329.890m
CD= 1307.615m
AD= 1182.419m
CD=1307.613m
CE=1190.228m
EH= 1210.978m
EF= 1552.434m
𝐶𝐸 = 1190.138m
CI = 796.061m
HG= 2076.658m
DF= 1416.232m
HJ=4690.414m
GJ= 4451.576m
GK= 4200.416m
𝐽𝐾 = 2249.649m
. Cálculos de azimut.- Estos cálculos se hicieron por la regla de la nemónica
𝐴𝑍𝐴𝐵 = 25°00ˡ00ˡˡ 𝐴𝑍𝐵𝐶 = 135°00ˡ59ˡˡ 𝐴𝑍𝐶𝐷 = 221°55ˡ52ˡˡ
𝐴𝑍𝐴𝐷 = 297°30ˡ18ˡˡ
𝐴𝑍𝐶𝐼 = 76°58ˡ48ˡˡ 𝐴𝑍𝐼𝐻 = 107°16ˡ17ˡˡ 𝐴𝑍𝐻𝐺 = 160°22ˡ11ˡˡ
𝐴𝑍𝐺𝐹 = 263°34ˡ55ˡˡ 𝐴𝑍𝐹𝐷 = 327°32ˡ26ˡˡ 𝐴𝑍𝐶𝐸 = 320°15ˡ17ˡˡ
𝐴𝑍𝐻𝐽 = 89°50ˡ54ˡˡ 𝐴𝑍𝐽𝐾 = 182°23ˡ54ˡˡ
13
TRABAJO DE TOPOGRAFÍA II
.Cálculo de coordenadas
𝑋𝐴 = 170000m 𝑌𝐴= 8500700m
𝑌𝐵 = 𝑌𝐴 + ABcos(𝐴𝑍𝐴𝐵) 𝑌𝐵 = 8501787.569 m
𝑋𝐵 = 𝑋𝐴 + ABsen(𝐴𝑍𝐴𝐵) 𝑋𝐵 = 170507.142m
𝑌𝐶 = 𝑋𝐵 + BCsen(𝐴𝑍𝐵𝑐) 𝑌𝑐 = 8501263.679m
𝑌𝐶 = 𝑌𝐵 + BCcos(𝐴𝑍𝐵𝑐) 𝑋𝑐 = 171740.367m
𝑌𝐷 = 𝑌𝐶 + CDcos(𝐴𝑍𝐶𝐷) 𝑌𝐷 = 8500153.928m
𝑋𝐷 = 𝑋𝐶 + CDsen(𝐴𝑍𝐶𝐷) 𝑋𝐷 =171048.771m
𝑌𝐸 = 𝑌𝐶 + CEcos(𝐴𝑍𝐶𝐸) 𝑌𝐸 = 8502178.839m
𝑋𝐸 = 𝑋𝐶 + CEsen(𝐴𝑍𝐶𝐸) 𝑋𝐸 = 170979.510m
𝑌𝐼 = 𝑌𝐶 + CIcos(𝐴𝑍𝐶𝐼) 𝑌𝐼 = 8501443.025m
𝑋𝐼 = 𝑋𝐶 + CIsen(𝐴𝑍𝐶𝐼) 𝑋𝐼 =172515.962m
𝑌𝐻 = 𝑌𝐼 + IHsen(𝐴𝑍𝐼𝐻) 𝑌𝐻 = 8501171.300m
𝑋𝐻 = 𝑋𝐼 + IHcos(𝐴𝑍𝐼𝐻) 𝑋𝐻 =173389.908m
𝑌𝐺 = 𝑌𝐻 + HGsen(𝐴𝑍𝐻𝐺) 𝑌𝐺 = 8499215.337m
𝑋𝐺 = 𝑋𝐻 + HGcos(𝐴𝑍𝐻𝐺) 𝑋𝐺 =174087.560m
𝑌𝐹 = 𝑌𝐺 + FGcos(𝐴𝑍𝐹𝐺) 𝑌𝐹 = 8498959.029 m
𝑋𝐹 = 𝑋𝐺 + FGsen(𝐴𝑍𝐹𝐺) 𝑋𝐹 =171809.007m
14
TRABAJO DE TOPOGRAFÍA II
𝑌𝐽 = 𝑌𝐻 + HJsen(𝐴𝑍𝐻𝐽) 𝑌𝐽 = 8501183.716m
𝑋𝐽 = 𝑋𝐻 + HJcos(𝐴𝑍𝐻𝐽) 𝑋𝐽 = 178080.306m
𝑌𝐾 = 𝑌𝐽 + HJcos(𝐴𝑍𝐽𝐾) 𝑌𝑘 =8498936.037m
𝑋𝐾 = 𝑋𝐽 + HJsen(𝐴𝑍𝐽𝐾) 𝑋𝑘 = 177986.165m
PUNTO NORTE ESTE A 8500700.000m 170000.000m B 8501787.569 m 170507.142m C 8501263.679m 171740.367m D 8500153.928m 171048.771m E 8502178.839m 170979.510m F 8498959.029 m 171809.007m G 8499215.337m 174087.560m H 8501171.300m 173389.908m I 8501443.025m 172515.962m J 8501183.716m 178080.306m K 8501183.716m 177986.165m A 8500700.000m 170000.000m
α1 = 63° 11ˡ 14ˡˡ + 1” α4 = 69° 52ˡ 46ˡˡ + 3” α7 = 64° 57ˡ 41ˡˡ + 2”
α2 = 60° 47ˡ 43ˡˡ + 1” α5 = 61° 54ˡ 25ˡˡ + 3” α8 = 63° 43ˡ 25ˡˡ + 1”
α3 = 56° 00ˡ 01ˡˡ+1” α6 = 48° 12ˡ 41ˡˡ + 2” α9 = 51° 18ˡ 50ˡˡ + 1”
∑= 179°59ˡ57ˡˡ ∑= 179°59ˡ52ˡˡ ∑= 179°59ˡ56ˡˡ
ERROR: 3 ERROR: 8 ERROR: 4
15
TRABAJO DE TOPOGRAFÍA II
CONDICIÓN TRIGONOMÉTRICA:
CAMINO I
𝐵𝐶
𝑠𝑖𝑛𝛼2=
𝐴𝐶
𝑠𝑖𝑛𝛼1 𝐴𝐸 =
𝐵𝐶 𝑠𝑖𝑛𝛼1 𝑠𝑖𝑛𝛼4
𝑠𝑖𝑛𝛼2 𝑠𝑖𝑛𝛼6 ………………………… I
𝐴𝐶
𝑠𝑖𝑛𝛼6=
𝐴𝐸
𝑠𝑖𝑛𝛼4
CAMINO II
𝐷𝐸
𝑠𝑖𝑛𝛼9=
𝐶𝐸
𝑠𝑖𝑛𝛼8 𝐴𝐸 =
𝐷𝐸 𝑠𝑖𝑛𝛼4 𝑠𝑖𝑛𝛼8
𝑠𝑖𝑛𝛼9 𝑠𝑖𝑛𝛼5 ………………………… II
𝐶𝐸
𝑠𝑖𝑛𝛼5=
𝐴𝐸
𝑠𝑖𝑛𝛼4
De I y II se tiene:
1 = 𝐵𝐶 𝑠𝑖𝑛𝛼1𝑠𝑖𝑛𝛼1𝑠𝑖𝑛𝛼9
𝐷𝐸 𝑠𝑖𝑛𝛼2𝑠𝑖𝑛𝛼6𝑠𝑒𝑛8 = ∆
eˡˡ = Log∆
∑ i FORMULA NUEVA
1 = 10,6415x10−7 1 = 11,7994x10−7 1 = 11,2387x10−7
1 = 18,8176x10−7 1 = 10,3952x10−7 1 = 16,8598x10−7
1 = 79,722x10−7
eˡˡ = 7,1323x10−6
7,9722x10−6 = 0.89ˡˡ
Esta corrección restamos a 𝛼1, 𝛼5 y 𝛼9 y sumamos a 𝛼2.
Ángulos corregidos
α1 = 63° 11ˡ 14ˡˡ α4 = 69° 52ˡ 49ˡˡ α7 = 64° 57ˡ 43ˡˡ
α2 = 60° 47ˡ 45ˡˡ α5 = 61° 54ˡ 27ˡˡ α8 = 63° 43ˡ 27ˡˡ
α3 = 56° 01ˡ 01ˡˡ α6 = 48° 12ˡ 44ˡˡ α9 = 51° 18ˡ 50ˡˡ
Calculo del lado AE:
𝐴𝐸 = 388.918 𝑠𝑖𝑛𝛼1 𝑠𝑖𝑛𝛼4
𝑠𝑖𝑛𝛼2 𝑠𝑖𝑛𝛼6 = 500.771 m
𝐴𝐸 = 409.573 𝑠𝑖𝑛𝛼4 𝑠𝑖𝑛𝛼8
𝑠𝑖𝑛𝛼9 𝑠𝑖𝑛𝛼5 = 500.772 m