MAXIMUM SEPARATION OF CHORD AND ELLIPTIC ARC 



In Figure 14, H^ is the maximum separation between the great elliptic arc and the chord. 

 As shown, this occurs when the tangent to the ellipse is parallel to the chord. Also when 

 this occurs the center of the ellipse, the midpoint of the chord, and the point P on the curve 

 are collinear, [10]. Hence the geographic coordinates of the point P can be found from the 

 intersection of the meridian through Q and the plane of the great elliptic section. 

 The coordinates of Q, the midpoint of the chord Q1Q2, are 

 f (a/2) (cos02 cos AA + cos d^) 

 Q < (a/2) (cos ^2 sin A A) 

 ' (b/2) (sin (9, + sin d^) 

 and the meridian through Q has the equation (cos 0^ sin AA) x - (cos 0, + cos 62003 A\)y = 0. (53) 

 The equation to the plane of the great elliptic arc in terms of parametric latitude is 



Ax + By + Cz = 0, (54) 



A = b tan 01 sin A A, B = b (tan 62 - tan 6^ cos A A), C = - a sin A A 

 (Compare equation (14), where it is in terms of geodetic latitude cfe). Now the point P 

 (a cos 6 cos A, a cos 6 sin A, b sin 6) on the the ellipsoid must satisfy both equations (53) 

 and (54) if it is to be the required point P on the great elliptic arc. This leads to the 

 equations cos 62 sin A A cos A — (cos 6^ + cos 62 cos A A) sin A = 0, 



A cosA+ B sin A+ C tan = 0, (55) 



where A, B, C are those of equation (54). 

 Solving (55) for A and d find, 



( A = arc tan [(cos ^jsin AA)/(cos ^jCos A A + cos di)]. 



6 = arc tan 



d = arc tan 



(tan di sin A A) cos A + (tan O^— tan 0, cos A A) sin A 



sin A A 

 tan 02 sin A + tan di sin (A A - A) 



(56) 



sin A A 



=arc tan [(sin d^ + sin 0j)/(cos^0i + cos^^j + 2 cos 0, cos 62 cos AA)''^^]. 

 Pe have seen that 



cos (di + dj) = sin 61 sin 62 + cos 0, cos 62 cos AA 

 sin 01 = sin O^ cos di, sin 02= sin d^ cos dj 

 vhence we can express 



cos^0i + cos^02 + 2cos 01 cos 02 cos A A = [l + cos(di+d2)][2-sin^0o!l + cos(di-d2)!] , 

 (sin 01 + sin 0j)^ = sin^0o [l + cos (dj + dj)] [1 + cos (d, - d2) ] 



(57) 



55 



