DERIVATION OF THE EQUATIONS TO SPHERICAL HYPERBOLAS 



Having established a rectangular spherical coordinate system on a great circle base line, 



we are now in a position to develop the equations of spherical hyperbolas referred to our 



rectangular system. Referring again to Figure 5, we restrict the point Q{6, A) or 0(x,y) to the 



locus defined by demanding that the distances ajand (jj from the points Qj and Q, respectively 



satisfy the condition 



a, - a, = 2c/e = 2a 



(27) 

 2c = S. - Sj , 



where as before Si, Sj are the distances of Qj, Qj respectively from (^o , X.q); e is a number 



such that e > 1. 



From the spherical triangles MQQi, MQQj one has 



cos (72 = f^os r cos c + sin r sin c cos a 



cos CTj = cos r cos c — sin r sin c cos a (28) 



Adding and substracting respective members of (28) obtain 



cos CTi+ cos 02= 2 cos r cos c 



cos CT, — cos ffj = ~ 2 sin r sin c cos a (29) 



By well known trigonometric identities and condition (27), equations (29) may be 

 written 



cos ff 1+ cos CTj = 2 cos /4(cti+ (72) cos Viia I- ff 2) = 2 cos /^(ai + cfj) cos a = 2(cos r) (cos c), 



cos (Ti- cos 02- 2 sin ^^(CTi + aj) sin Yiioi -ffj) = 2 sin K(a, + cr2) sin a = -2(sin r) (sin c) cos a, 

 or cos /4(<7i + 02) = cos r cos c/cos a, 



sin H (cTi + CT2) = sin r sin c cos a/sin a. 



Squaring and adding respective members of (30), get 



(cosM (cos^c/cos^a) + (sin^r cos^a) (sin^c/sin^a) = 1. (31) 



Now in (31) place cos^r = 1/(1 + tanM, 



sinV = tan^r/(l + tan^r), whence (31) may be written 



tan^a (cos^a - cos^c) tan^a (sin c - sin a) 



(30) 



(32) 



sinccosa-sina sinccosa~sina 



Now (32) is the polar form of the equation to the spherical hyperbola. 

 From conversion formulas (15) we have 

 tan^r = (sin^x + sin^y)/(l - sin^x - sin^y), 

 cos^a = sin^x/(sin^x + sin^y) (33) 



31 



