For equally spaced intervals along the great circle track, for instance in 100 nautical mile 



intervals, let S = S,+ lOOK, K = 1, 2, 3, , n. With these values of S one computes 



successively corresponding values oi 6 , X , and a from 



sin 6'— sin d^, cos S, tan iX^ — \ ') = tan S/cos do , tan a'= cot 6^ /sin S 

 and checks by means of sin 0' tan (Ao- A') • tan a'= 1 . 



PARALLELS AT A GIVEN DISTANCE FROM THE GREAT CIRCLE TRACK 



To compute the coordinates (0p, Ap) and {9p', Ap') of points at a given distance s from a given 

 great circle track and symmetric with respect to it one computes (see Figure 3): 



sin 01 = A cos S ± B when k = p, use + sign 



\ r^ . c / k = p ' , use - sign 



sin (Ag— A|^) = C sin S/cos 9i 



S and du are the same as given under the great circle track formulas above and A = C sin 0^, 

 B = cos do sin s, C = cos s. The computations may be checked by 

 cos 2s = sin &p sin dp' + cos 9p cos 0p'cos (Ap'— Ap). 



SPHERICAL RECTANGULAR COORDINATE SYSTEM WITH A GREAT CIRCLE BASE LINE AS 

 AN AXIS 



It is assumed that the base line has been established, that is the coordinates {do, Ao) 

 of the vertex of the great circle base line have been computed from the coordinates of two given 

 points 0,(^1, Aj), 02(^2, A2), see Figures 2 and 5. 



Formulas for computing y, S, x from d and A 



sin y = cos do sin d - sin do cos d cos (Ao - A) 



sin do cos d sin (Aq— A) cos d sin (Aq — A) 



tan S = — :- 



sin d — cos do sin y sin do sin d + cos do cos d cos (Ao - X) 



sin X = sin (S — SJ cos y 



Formulas for computing S, d, X from x and y 

 Lei C = cos y, D = sin y, E = sin x, A = C sin do, B = D cos ^ , then 



S = arc sin (E/C) + Sj 

 d = arc sin (A cos S + B) 

 A = Ao — arc sin (C sin S/cos d) 



