DEVELOPMENT 

 SECTION 2. SPHERICAL RECTANGULAR COORDINATE SYSTEM; LOCI 



THE GREAT CIRCLE TRACK AS DETERMINED BY THE GEOGRAPHICAL COORDINATES OF 

 TWO GIVEN POINTS ON THE AUXILIARY SPHERE 



In figure 2, the two given points are Qi(^i, Aj), QjC^j, A 2). The great circle track is then 



determined from the spherical triangle PQiQj. In order to simplify the computations and to have 



well balanced triangles from which to compute, one finds the point 0(6o,\o) where the great circle 



QiQj is orthogonal to a meridian X„. One then works from the right spherical triangle POQ'by 



adding or subtracting increments of distance from Sj = OQi to get the distance S. One always has 



then a strong right triangle POQ'from which to compute the latitude, longitude and azimuth a 



of the point Q'(d',\') on the base line Q,Q2 • 



DERIVATION OF FORMULAE 

 From right spherical triangle POQ ' 



cos (Ao-A') = tan(— -00 >ot( - -6*')= cot ^o tan d' (1) 



2 2 



If the points Qi and Q2 satisfy (1), we have by substituting their coordinates in (1) 

 cos ( Aq- Ai ) = cot da tan 6^ , (2) 



cos ( A(, — A2) = cot do tan 62 



By forming the ratios of (2), expanding cos (A,,— Ai) and cos (Aq— Aj), dividing the left 

 member numerator and denominator by cos A^ one derives the formula 



tan 02 cos Ai — tan 9j cos A 2 , , 



tan Aq = ■_ ■_ • \i) 



tan 0, sin A2 — tan 62 sin Aj 



Equations (2) may be written as 



cot do = cot di cos ( Ao — A ,) = cot 02 cos ( Aq— Aj) (4) 



From right spherical triangle POQ 'one has also 



sin( — - do ) cos do 

 sin a'= 2 = , (5) 



sin( — — 0' ) cos d' 



' tan S c /I' i^\ 



cos a = = tan b tan d , (O) 



tan(- -d') 



^2 ^ 



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