Y = 



Great Elliptic Arc Distance 



s/a - (d. + dj) - % k' [(di + d,) - sin (d, + d^) cos (d, - d^)] 



- (1/128) k' [6(di + dj) - 8 sin (d. + d,) cos (d^ - dj) + sin 2(di + dj) cos 2{A^ - dj)] 



- (1/1536) k' [30(di + dj) - 45 sin (d, + dj) cos (d, - dj) + 9 sin 2(d, + dj) cos 2(di - d,) 



- sin 3(di + dj) cos 3(di - dj)] 



Where in terms of geodetic latitude <;6, 



k = (e\/ 1 - e /a) N^ sin (^o > ^j = arc cos (N, sin 0i/No sin <;6o)> 



dj = arc cos (N2 sin (^j/Nj sin 0^) 



sin (;6o = [J/(J + si"^ AA)]'/S J - tan^^i + tan^c^j - 2 tan 0^ tan <^2 cos AA, 



and in terms of parametric latitude Q 



k = e sin Q^, dj = arc cos (sin 0,/sin ^0), dj = arc cos (sin (9 j/sin Q^ 



sin 00 = [F/(F + sin' AA)]'/', F = tan'(9i + tan'efj - 2 tan 0, tan Q^ cos AA. 



Also in terms of parametric latitude Q, great ellipticarc distance 



d - (eV8) (Xd - Y sin d) ~| 



- (eV512) [(6d-sin2d) X' - 8 (sin d) XY + 2 (sin 2d) Y'] 



_ - (eVl2288) [3(10d - 3 sin 2d) X' - 3(15 sin d - sin 3d) X'Y + 18(sin 2d) XY'-4(sin 3d) Y'jJ 



(sin 0, + sin Q^'^ (sin Q - sin Q^^, 



re X = + 



1 + cos d 1 - cos d 



(sin 01+ sin Q^ (sin Q^ - sin Q^'^ , d = d,- dj, where d,, A^ are spherical distances from Pi(0i, AJ, 



1 + cos d 1 - cos d 



Pj((9j, Aj) to the vertex Po(0o, ^o)- 



NOTE: If e' " 2f, the higher order terms in f then ignored, this becomes the so-called Andoyer-Lambert 

 approximation in terms of parametric latitude. 



GEODESIC IN TERMS OF GREAT ELLIPTIC ARC, IN GEODETIC LATITUDE WITH SECOND ORDER 



TERMS IN THE FLATTENING 



Given the points Pi( ^1, A J , V^k^i, Aj) on the reference ellipsoid, Pj west of Pi, west longitudes 



considered positive. 



With <^^ = Y.A4>^ + 0,), A<^„ = y2(</,, - <^i), AA = A,- A., AAn, = KAA, 



Let k = sin </> ^ cos A0 j^, K = sin A(;6 ^ cos „, , 

 H = cos A(;6jjj - sin'<^jj| = cos'^u^ - sin A<;6^ 



L = sin'AvSj^ + H sin'AAn^ = sln'(d/2), 1 - L = cos'(d/2), cos d = 1 - 2L, 

 t = sin'd = 4L(1 - L), U = 2kV(l - L), V - 2KVL; X-U + V, Y-U-V, 



