COLLECTED FORMULAE 



NEW LATITUDE FORMULAS 



If 6 is the geocentric latitude of a point P(acos0, a sin0) on the auxiliary sphere, then the 

 corresponding geodetic latitude ^ of P at an altitude h above the ellipsoid of reference as shown 

 in Figure 1, is given by 



sin A4> = sm{4i-d) = (e'/2a) Nsin 20= (e' sin (^cos 0) /(1-e' sin'0)V^ 

 = Cjsin 2(f) — Cjsin 4(^ + Cjsin 6ci — C4sin 8(f>, 

 c, = (e72) + (eV8) + (15eV256) + (35er/1024) 

 c^ = (eVl6) + (3er/64) + (35er/1024), 

 C3 = (3eV256) + (15eV1024), 

 c, = 5eV2048 

 With the same coefficients, 



cfi- 6= ^<fl (radians) = (c^ + CiV8) sin 2^^ - (c^ + c/c^A) sin 40 + (cj - CiV24) sin 60 

 A0 (seconds) = (206,264.8062). A0 (radians). 



To express A0 in terms of d, we have 

 tan0 = tan 6 + (e^/a cos(?) N sin0 



= tan6l+ (eVcos^) sin 0/(1 - e'sin '0)'/', 

 which, when expanded by the Lagrange expansion formula gives 

 A0 = 0— 6 = Cisin20+ C2sin40+ C3sin 66 + C4sin80 

 ci = (eV2) + (eV8) + (lle^/256) + (31eVl024) 

 cj = (3eVl6) + (5eV64) + (25eVl024) 

 C3 = (77eV768) + (59eVl024), 

 C4 = 127eV2048 

 The distance h is given by 

 h/a = cos A0 - a/N = cos A0 - (1 - e'sin'0)V'' 



= (l-e='sin'0)-'A([l-e'sinV(l + e'cos'0)]'/'- 1 + e'sin'0} 

 h = a(di — d2Cos 20 + djCos 40 — d4Cos 60 + d; cos 80) 



d, = (eV4) - (eV64) - (3eV256) - (233eVl6384) 



d^ ^ (eV4) + (eVl6) + (7eV512) + (3eV2048) 



dj = (5eV64) + (lleV256) + (115eV4096) 



d4 = (9eV512) + (37eV2048) 



d, =53eVl6384 



