SUMMARY OF LATITUDE FORMULAE 



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

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

 shown in figure 1, is given by 



sin A^ = sin {(f, - ff) = (eV2a) N sin 20 = (e^sin <^ cos 0)/^ 1 - e^sin^^ 



= c, sin 20 — Cj sin 40 + C3 sin 60 — c, sin 80, (49) 



Ci= eV2 + eV8 + 15e' /256 + 35eVl024, 

 cj = eVl6 + 3eV64 + 35eVl024 

 C3 = 3eV256 + 15eVl024, c, = 5eV2048 

 e = eccentricity of the meridian ellipse. 



With the same coefficients as (49), we have 



A0 (radians) = (cj + Ci^ /8) sin 20 - (cj + Cj) sin 4 +(c3 ) sin 60 (50) 



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and in seconds 



(51) 

 A0(seconds) = (206,264.8062) [ (c^ + c,V8) sin 20 - (cj +c,=' Cj/4) sin 40 + (c^ - 0^724) sin 60] . 



To express A0 in terms of 6, instead of 0, we have the relation 



tan = tan 6 + (e^/a cos 6) N sin 



Which may be expanded by use of the Lagrange expansion formula to give 



A0 = - = C, sin 26 + C^ sin 40 + C3 sin 66 + C^ sin 86 

 C, = e72 + e78 + lleV256 + 31eVl024, (52) 



Cj = 3eVl6 + 5eV64 + 25eVl024, 

 C3 = 77eV768 + 59eVl024, C, = 127eV2048. 



For checks within 0.001 second, (52) may be written A0 (seconds) = (206,264.8062) 

 (C . sin 20 + Cj sin 40 + C3 sin 6(9) (53) 



with C,, Cj, C3 the same as in (52). 



h/a = cos A0 - a/N = (1 - e^ sin^-'^^'Ul - e^ sin^ (1 + e' cos' 0)]'/^- 1 + e' sin' } 



h = a(di -dj cos 20 + dj cos 40 - d4 cos 60 + d^ cos 80) (54) 



d. = e74 - e764 - 3e7256 - 233e7l6,384 



dj = e74 + e7l6 + 7e7512 + 3e72048 0<h<a-b 



dj = 5e764 + lle7256 + 115e74096 



d^ = 9e7512 + 37e72048, d^ = 53e7l6,384 



a = radius of the auxiliary sphere (semimajor axis of the reference ellipsoid). 



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