Loxodromic Distances on the Terrestrial Spheroid

Abstract

This paper is concerned primarily with an investigation of the loxodromic distance between two points on the spheroid. This is derived by a modification of the meridional parts method in terms of the track angle and the difference of latitude reduced by a correction which is a function of the latitudes concerned.Since the method is unsuitable when the track angle is near to 90°, the known formula for the mid-latitude correction is examined and an alternative method suggested for use in this case.The results obtained are compared in a practical example with those found by the usual methods, and finally means are proposed by which the refinement described above can be incorporated into current plotting practice on aeronautical Mercator charts.

Practical Rhumb Line Calculations on the Spheroid

“…Calculate M x , M Y , m x and m Y using formulae (1) and (2). Solve for C and S from tan C = AX/AM and 5…”

“…<j> Y may then be obtained from formula (3) which has been derived empirically by reversing the series formula (2). However, it is not essential to have formula (3) in a computer program because latitude can be derived by successive approximations using an initial value of <f> = m/ Go etc.…”

“…and Q^will be defined and explained in the following section. There are more than sufficient terms in formulae (2) and 3for most marine applications. The user may therefore choose the number of terms that are sufficient for the application under consideration.…”

“…However, it is not essential to have formula (3) in a computer program because latitude can be derived by successive approximations using an initial value of <f> = m/ Go etc. in formula (2). Three iterations are sufficient for most marine navigation applications.…”

Exploration with a Computer on Rhumb-line Sailing

“…Differences from a precise result arise from the method used to allow for the ellipticity of the Earth and also from the fact that the distance so calculated is not given in Admiralty nautical miles (6080 ft.). The nature of these differences is adequately explained in previous papers by Sadler, 1 and Williams, 2 where precise methods are given.…”