PREFACE 



Early in World War II, the U. S. Navy Hydrographic Office began publishing charts and tables 

 for the new loran system of long-range radio navigation. Loran and similar systems make use of 

 radio waves to measure earth distances or distance-differences (hyperbolas) for positioning ships 

 or aircraft at long ranges from the shore transmitter stations. The computation of these naviga- 

 tional lines of positioning is a problem in geodesy. Because of the irregularities of the shape of 

 the actual earth over which radio waves travel, geodesists are forced to use mathematical models 

 that approximate the shape of the earth when computing navigational lines of position. 



Various models and co-ordinate systems have been used in making loran-type computations, 

 which were originally done by desk calculators within limits of accuracy compatible with the early 

 navigation systems. Now, however, improved system accuracies and better information of the 

 figure of the earth have made necessary a re-examination of the mathematical formulas to ensure 

 their adequacy at very long ranges. 



The inverse distance formula used in loran computations is actually the so-called Andoyer- 

 Lambert approximation and is the expansion of the geodesic arc length between two points on the 

 reference ellipsoid to first order in the flattening. There are two simple and very similar forms of 

 the approximation, one in terms of geodetic latitude and the other in terms of parametric latitude. 

 The U. S. Naval Oceanographic Office uses the latter which requires a conversion from geodetic 

 latitude. While the parametric form gives slightly more accurate distance computations, the 

 objective of this study was to determine whether the latitude conversions are justified and to 

 investigate the second-order terms in the expansions and their contribution to the accuracy of the 

 computations. 



It was the conclusion of the study that the parametric formulas which have been used are in 

 fact adequate to meet present operational requirements but that the conversion to parametric 

 latitude is not necessary. In anticipation of future requirements, the geodetic formulas were 

 extended to give geodesic distances and azimuths between any two points on the reference el- 

 lipsoid to uncertanties of less than a meter and a second respectively, out to ranges of 6000 

 miles. 



During the investigation, formulas were developed for the particular quantities involved and 

 were transformed in terms of particular computational parameters. Some associated useful geo- 

 metrical quantities were included relative to distance computations: chord distance, the dip of 

 the chord, the maximum separation distance between chord and arc (surface), and the geographic 

 position of the point where maximum separation occurs. Some of these relationships can be found 



