General series expansion and mathematical analysis of three auxiliary latitudes of meridian arc length

The calculation of meridian arc length is typically expressed as a series expansion of geodesic latitude. However, during the course of our research into map projection, we discovered that for different types of map projections, it is often necessary to introduce three types of auxiliary latitude: equal angle latitude, equal distance latitude and equal area latitude. Mathematica was employed to derive the relationship between these latitudes and the arc length of the meridian, with the third flattening <i>n</i> being used instead of the first eccentricity to re-derive their coefficients. The analysis indicates that the formula for the length of a meridian expressed in terms of equiangular latitude is more accurate and efficient than that expressed in terms of equidistant area latitude. Consequently, it is more appropriate to use the formula for the length of a meridian expressed in terms of equiangular latitude in the related fields of geodesy, map projection and so on.