let k = sin 6^ cos M^, K = sin Ad^ cos 6^, 

 H = cos'Ae^ - sin'eij^ = cos'(9^ - sin' Al9^, 



L = sin'Aeij^ + H sin' AA^ = sin'd/2, 1-L = cos'd/2, 



cos d = 1 - 2L, h = sin'd = 4L(1 - L), U = 2kV(l - L), 



V = 2r/L, X=U + V, Y = U-V, 



T = d/sin d = 1 + (l/6)h + (3/40)h' + (5/112)h^ + (35/1152)h^ + (63/2816)h= + , 



E„ - - 2 cos d, Do = 4TS Ao = -DoEo, B„ - -2D„, C^ = T - H(A„ + E„), 



S ■= a sin d [T - (f/4) (TX -Y) + (fV64) (AoX + BoY + C„X' + D„XY + E„Y')] 



sin («!+ ai) = (K sin AA)/L, sin {a^ - a,) = (k sin AA)/(l - L) 



'/2(Sa2 + 5a,) = - (f/2) TH sin (a, + a^) 



'AiSa, - Sa,) = - (f/2) TH sin {a, - a.) 



ai_2 = Oi + ha-i , 02-1 = a2+ Sa^ 



Additional check formulae 



(sin di + sin O2Y (sin 0, - sin ^j)' 



X = + = 2 sin'^o = 2F/(F + sin'AA) 



1 + cos d 1 - cos d 



(sin 0j + sin d^)' (sin d^ - sin ^j)' 



Y = = 2 sin ^ocos (d, + d^) 



1 + cos d 1 — cos d 



F = tan'^i + tan ^62 - 2 tan 6^ tan d^ cos AA 



cos (d. + d^ = Y/X, 1 + cos d = 8kV(X + Y), 1 - cos d = 8KV(X - Y), 



/ k' K'\ /k' K'\ 

 cos d = 4 - , 4 + =1. 



yx+Y x-Yy yx+Y x-Yy 



NOTE: If the second order term is ignored, the resulting equations are the equivalent of the so-called 

 Andoyer-Lambert approximation in terms of parametric latitude. 



TRANSFORMATIONS: GEODETIC TO PARAMETRIC — PARAMETRIC TO GEODETIC 



If primed quantities denote those in geodetic latitude, then the transformation equations are: 



d ' = d - (f/2) Y sin d + (f7l6) [4Y(X-3) sin d + (2Y' - X') sin 2d], 

 sin d'=sin d - (f/4) Y sin 2d 

 X'=X[1 +f (2-X)] 

 Y '= Y[l + f (2 - X)] + (f/2) (X' - Y') cos d 



d = d'+(f/2) Y'sin d'+(fVl6) [4Y'(X'-l)sind'+(2Y''-X'')sin2d1 

 sin d = sin d'+ (f/4) Y'sin 2d' 

 X =X'[1 -f(2-X')] 

 Y=Y'[1 - f(2 - X')] - (f/2) (X"-Y") cos d' 



10 



