PARALLELS AT A GIVEN DISTANCE FROM A GREAT CIRCLE TRACK 



In Figure 3, the basic great circle track determined by Q, (flj, Aj), Q^ iO^, Aj) is the same 

 and the point 0((9o, Aq) is the same — (vertex of the great circle track). The point P' is the pole 

 of the great circle determined by Qi, Q2. The angle at P ' of the spherical triangle P 'PQ is the 

 distance S = OQ along the great circle track. If p and p ' are points on the parallels at a distance 

 s from the great circle track, then the coordinates of p and p' can be computed from the two 

 spherical triangles PP'p, PP p ', (Figure 4). 



2 P 



■?7-(A„-A ) 

 P 



Figure 4 



From these triangles one has 



sin dp = cos do sin s + sin do cos s cos S 



sin 6p' = — cos do sin s + sin do cos s cos S 



cos s cos ^p cos s c 



sin(A(|-Ap) sin S sin(Ao-Ap') sin S 



From (11) and (12) one may write 



sin % = A cos S + B 



sin (Aq — A ]^) = C sin S/ cos 0|^ 

 where A = sin do cos s, B = cos dg sin s, C = cos s. 



A, B, C are constants for a given s. When k = p, the + sign is used in the first of 

 equations (13). When k = p ', the — sign is used. 



The computations may be checked as before by means of the equation 

 cos 2s = sin dp sin dp'+ cos dp cos 0p' cos (Ap'— Ap). 



(11) 

 (12) 



(13) 



27 



