STANDARD LATITUDE FORMULAS 



The three latitudes usually associated with the auxiliary sphere ellipsoid configuration as shown 

 in Figure 1, are the geocentric, parametric, and geodetic represented here by i//, 6, and (jAq 

 respectively and related through the equations 



tan i/i/tan 6 = tan 0/tan ^o = (1 -e^Y'^, 

 where e is the eccentricity of the meridian ellipse. The parametric latitude, 6, is also called here 

 the geocentric latitude of points on the auxiliary sphere. 



LATITUDES FOR CLARKE 1886 SPHEROID 



Series representations, accurate to 0.001 second, for the differences in 0, <^o> d, ifj are: 



A0 (seconds) = ^ - = 699 :'2540 sin 2<^ - 15936 sin 4<;6 + 0"0004 sin 60 



A0 (seconds) = - = 699':2520 sin 2d + l':7769 sin 4 + 0':0064 sin 66 



A9g (seconds) = 4> - 4>o -^ 349':0318 sin 26 + l"A796 sin 4(9 + 0':0061 sin 66 



h (meters) = 10,788.3852 - 10,811.2646 cos 2(p + 22.9147 cos 40 - 0.0350 cos 60 



0^ - i/' = 700':4385 sin 20^, - r:i893 sin 40^ + 0':0027 sin 60^ 



0Q - ^ = 700':4385 sin 20 + 1 :'1893 sin 40 + 0':0027 sin 60 



0^ - = 350".2202 sin 20^ - 0':2973 sin 40^ + 0':0003 sin 60^ 



0^ - = 35012202 sin 2d + 0':2973 sin 40 + O'lOOOS sin 66 



(9 - = 350':2202 sin 26 - 0:'2973 sin 40 + O'lOOOS sin 66 



6-i/f^ 350';2202 sin 20 + 0':2973 sin 40 + 0';0003 sin 60 



GREAT CIRCLE TRACK FORMULAS 

 First compute A and 6 from 



tan 62 cos A J— tan 61 cos A, 



tan An = 



tan 61 sin A2— tan 62 sin A , 

 cot 00 = cot di cos (A^— A,) = cot 62 cos (Ao - Aj). (See Figure 2). 

 Then compute Oj and 02 from 



cos 00 _ cos 00 



sin a, = , sin Oj = ^^—^— 



cos 01 cos 02 



Next compute Sj and Sj from 



tan Si = cos a 1 cot 0j , tan Sj = cos a 2 cot 02 



The computations for a^, aj , Sj and S2 are checked by 



cos (Aj — Ai) = cos a 1 cos Oj + sin a, sin Oa cos (Sj - Sj) 



