MATHEMATICAL MODELS 



FOR 



NAVIGATION SYSTEMS 



OVERALL SUMMARY OF INVESTIGATIONS 



Latitude 



A loran station positioned on the auxiliary sphere of the ellipsoid of reference has as its 

 geodetic latitude the angle at the equator made by that normal to the meridian which passes 

 through the station of the sphere. Its longitude will remain the same. See Figure 1, page 13. 

 Now this is analogous to the geodetic latitude of a subsatellite point, if the trajectory were 

 confined wholly to the surface of the auxiliary sphere. It is clearly not one of the three 

 commonly associated latitudes as shown in equation (1), page 12 . Actually the relationship 

 between geocentric latitude on the sphere and geodetic latitude on the ellipsoid is given by 

 equation (2), page 12 . From these one may find the maximum value of the difference, A<?i), and 

 the value of 0, the geodetic latitude, at which this maximum difference occurs, equations (3) — 

 (6), page 14. The expansions of S.cfi in series in terms of (fj were obtained and are given in 

 equations (7) — (20), pages 15, 16. 



For computation of <;6 as a function of 6 , the geocentric latitude, it was necessary to employ 

 the Lagrange expansion formula and the resulting expansion and formulas are given in equations 

 (21) — (33), pages 16 to 18. Development of the series expansions for the height, h, of the 

 auxiliary sphere above the ellipsoid is given in equations (43^— (48). See Figure 1, page 13 

 and pages 19,20 . A summary of latitude formulas and a bibliography of pertinent references 

 are included, pages 21 —22. 



The great circle track as determined by the geographical coordinates of two given points on the 

 auxiliary sphere. Parallels at a given distance from a great circle track . 



As shown in figure 2, page 24, the treatment is somewhat different than the usual one in 

 that one works from the vertex of the great circle or the point where the great circle is or- 

 thogonal to a meridian. This simplifies computations and provides well balanced triangles 

 from which to compute. It also facilitates the computations for parallels at a given distance 

 from a fixed great circle track as shown in Figures 3 and 4, pages 26 and 27. See also 

 equations (1) - (13), pages 23-27. 



