DEVELOPMENT 



SECTION 1. LATITUDE FORMULAE 



The auxiliary sphere, associated with an ellipsoid of reference, is the sphere tangent to the 

 spheroid along the equator. If it is desired to work on this sphere with formulae for conversion 

 to the spheroidal surface, then a correspondence between geocentric latitude d on the sphere and 

 geodetic latitude (j) on the ellipsoid is needed. Longitudes will be the same. 



Now there are three latitudes in geodetic usage associated with the auxiliary-sphere ellipsoid 

 configuration as shown in Figure 1. The 6 as shown, and which we shall call geocentric latitude, 

 is called the reduced or parametric latitude since it is the eccentric angle of the meridian ellipse. 

 The angle i/r , as shown, is called in geodetic nomenclature, the geocentric latitude since it is 

 the angle measured from the center of the ellipsoid to the point R on the meridian from the equator. 

 The angle ^o > as shown, is a geodetic latitude corresponding to 6. The three latitudes i/f, 6, cj)^, 

 are related through the equations 



tan ^fr = Vl - e"" tan = (1 - e^) tan cj)^ (l) 



or tan ip / tan = tan / tan 9S0 = \/l - e^ . 

 where e is the eccentricity of the meridian ellipse [l].* 



However, for working directly on the auxiliary sphere and transferring directly to the ellipsoid, 

 if 6 is the geocentric latitude of the point P (a cos 6, a sin 6) on the auxiliary sphere, then the 

 latitude actually corresponding on the spheroid is that found by dropping a perpendicular upon 

 the meridian ellipse from P meeting the meridian in Q as shown in Figure 1, the normal making 

 the angle cj) as shown with the equator. The distance PQ = h, and <f> are needed for the conversion 

 where 0<h<a-b, a and b the semimajor and semiminor axes of the spheroid. We now develop 

 the necessary conversion formulas between (^ and 6. 



The law of sines applied to triangles POT, POK of figure 1, yields 



Ne'sinc?!) ^ h + N ^ a Ne'cosq!) ^ h + N(l-e') ^ _a_ ^ 



sinA0 cos cos (^ sinA^S sin sin 



where N = a/Vl- e^sin^0 ; e, a are the eccentricity and equatorial radius of the reference 

 ellipsoid. (Ac^ ^ 4' ~ ^^- 



*[1] Bracketed numbers refer to the list of references at the end of the section. 



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