Las formulas de geodecia

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ESCUELA DE INGENIEROS MILITARESG
TECNOLOGÍA EN TOPOGRAFÍA

GEODESIA

EJERCICIO DE LEGENDRE

ING. MIGUEL AVILA

CS CUELLAR VALERO ADRIAN
COD.316301010

BOGOTÁ DC, 10 DE AGOSTO DE 2010
EJERCICIOS DE 6 AL 10 DE LEGENDRE

EJERCICIO # 6
P_(n ) =∑_(k=0)^n▒= ((-1)^(k ) (2*n-2*k)!)/(2^n*0!(n-k)!*(n-2*k)!) I^(n-2*k )

P6
n = 6
N = 6/2 =3

P6 ∑_(k=n)^1▒ ((〖-1)〗^0(2*6-2*0)!)/(2^6*0!(6-0)!(6-2*0)!) τ^(6-2*0 )+ ((〖-1)〗^1 (2*6-2*1)!)/(2^6*1!(6-1)!(6-2*1)!) τ^(6-2*1 )

+ ((〖-1)〗^2 (2*6-2*2)!)/(2^6*2!(6-2)!(6-2*2)!) τ^(6-2*2 )+ ((〖-1)〗^3 (2*6-2*3)!)/(2^6*3!(6-3)!(6-2*3)!) τ^(6-2*3 )

P6 = (1*12!)/(64*1*6!*6!) τ^(6 ) - 10!/(64*1*5!*4!) τ^(4 ) + (1*8!)/(64*2*4!*2!) τ^(2 ) - 6!/(64*6*3!*1) τ^(0 )

P6 = 231/16 τ^(6 ) - 315/16 τ^(4 ) + 105/16 τ^(2 ) - 5/16EJERCICIO NUMERO 7 COMPLETO CON P_nm

P_(n ) =∑_(k=0)^n ((-1)^(k ) (2*n-2*k)!)/(2^n*0!(n-k)!*(n-2*k)!) I^(n-2*k )

P_(7 ) =∑_(k=0)^3 ((-1)^(0 ) (2*7-2*0)!)/(2^7*0!(7-0)!*(7-2*0)!) I^(7-2*0 )- ((-1)^(1 ) (2*7-2*1)!)/(2^7*1!(7-1)!*(7-2*1)!) I^(7-2*1 )+ ((-1)^(2 ) (2*7-2*2)!)/(2^7*2!(7-2)!*(7-2*2)!) I^(7-2*2 )- ((-1)^(3 ) (2*7-2*3)!)/(2^7*3!(7-3)!*(7-2*3)!) I^(7-2*3 )P_(7 )=(1*(14)!)/(128*1*(7)!*(7)!) I^7-(1*(12)!)/(128*1*(6)!*(5)) I^(5 )+(1*(10)!)/(128*2!*(5)!(3)!) I^(3 )-(1*(8)!)/(128*3!*(4)!*(1)!) I^

P_(7 )=715/32 I^7-693/16 I^(5 )+315/16 I^(3 )-35/16 I

P_nm=〖 (1-τ^2 )〗^(3⁄2) * d^3/(dτ^3 )
n=7 m=3

P_73=〖 (1-τ^2 )〗^(3⁄2) * d^3/(dτ^3 ) [ 715/32 τ^7-693/16 τ^5 + 315/16 τ^3 -35/16 τ ]

1^RADERIVADA = 5005/32 τ^6-3465/16 τ^4 + 945/16 τ^2 -35/16

2^dA DERIVADA = 30030/32 τ^5-13860/16 τ^3 + 1890/16 τ

3^RA DERIVADA = 1501150/32 τ^4-41580/16 τ^2 + 1890/16

=750575/16 τ^4-10395/4 τ^2 + 945/8

P_73=〖 (1-τ^2 )〗^(3⁄2) *(750575/16 τ^4-10395/4 τ^2 + 945/8 )

P_73=〖 (1-〖COS〗^2 θ)〗^(3⁄2)*(750575/16 τ^4-10395/4 τ^2 + 945/8)

P_73=〖 (〖SEN〗^2θ)〗^(3⁄2)*(750575/16 〖COS〗^4 θ-10395/4 〖COS〗^2 θ + 945/8)

P_73=〖SEN〗^3 θ*(750575/16 〖COS〗^4 θ-10395/4 〖COS〗^2 θ + 945/8 )

VENADILLO φ=4^ο 〖44〗^, 〖36〗^" θ=〖90〗^ο-4^ο 〖44〗^, 〖36〗^"= θ=〖85〗^ο 〖15〗^, 〖24〗^"

〖SEN〗^3 〖85〗^ο 〖15〗^, 〖24〗^"*(46910,93*〖COS〗^4 〖85〗^ο 〖15〗^, 〖24〗^"-2598,75*〖COS〗^2 〖85〗^ο 〖15〗^, 〖24〗^" + 118,125)=

0,9897*(2,1934-17,7702+118,125)= 101,4919EJERCICIO #8

n=8 es par entonces N=n/2 N=8/2 N=4

Pn=∑_(K=0)^N (〖(-1)〗^(K ) (2*n-2*k)! )/(2^n k! (n-k)!(n-2*k)!) τ^(n-2*k)

P8=∑_(K=0)^4 ((-1)^(0 ) (2*8-2*0)! )/(2^8 0! (8-0)!(8-2*0)!) τ^(8-2*0)+((-1)^(1 ) (2*8-2*1)! )/(2^8 1! (8-1)!(8-2*1)!) τ^(8-2*1)+ (〖(-1)〗^(2 ) (2*8-2*2)! )/(2^8 2! (8-2)!(8-2*2)!)τ^(8-2*2)+(〖(-1)〗^(3 ) (2*8-2*3)! )/(2^8 3! (8-3)!(8-2*3)!) τ^(8-2*3)+(〖(-1)〗^(4 ) (2*8-2*4)! )/(2^8 4! (8-4)!(8-2*4)!) τ^(8-2*4)

P8=(1* 16! )/(256*1* 8!*8!) τ^8-(1* 14! )/(256* 1* 7!*6!) τ^6+(1*12! )/(256* 2!*6!*4!) τ^4-(1* 10! )/(256* 3!* 5!*2!) τ^2+(1 *8! )/(256 *4!*4!0!) τ^0

P8=(1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16)/(256*1*2*3*4*5*6*7*8*1*2*3*4*5*6*7*8) τ^8

-(1*2*3*4*5*6*7*8*9*10*11*12*13*14 )/(256* 1*2*3*4*5*6*7*1*2*3*4*5*6) τ^6

+(1*2*3*4*5*6*7*8*9*10*11*12 )/(256* 1*2*1*2*3*4*5*6*1*2*3*4) τ^4

-(1*2*3*4*5*6*7*8*9*10 )/(256* 1*2*3* 1*2*3*4*5*1*2) τ^2+(1*2*3*4*5*6*7*8 )/(256 *1*2*3*4*1*2*3*4)

P8=(77220 )/1536 τ^8-24024/256 τ^6+( 3860)/256 τ^4-(2520 )/256 τ^2+(70)/(256 )

P8=(6435 )/128 τ^8-( 3003)/32 τ^6+( 965)/64 τ^4-( 315)/32 τ^2+( 35 )/(128 )

P_8=(6435 )/128 τ^8-( 3003)/32 τ^6+( 965)/64 τ^4-( 315)/32 τ^2+( 35 )/(128 )


P_8=(6435 )/128 〖COS〗^8 θ -( 3003)/32 〖COS〗^6 θ + ( 965)/64 〖COS〗^4 θ -315/32 〖COS〗^2 θ + 35/128

EJERCICIO # 9

P9= n= 9 N= (N-1)/2=N=(9-1)/2 N=8/2 =4

P9=∑_(K=0)^4▒〖((-1)^0...
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