Approximation of a Great Circle by using a Circular Arc on a Mercator Chart

摘要

This paper describes George Biddell Airy's almost completely unknown method of approximating an orthodromic arc (great circle arc) using a circular arc in the normal aspect Mercator projection of a sphere. In addition, it is demonstrated that the centre of the circle can be defined in at least two different ways, which yields slightly different results. Airy's approach is built upon in this paper. The method of computing coordinates of Airy's circle arc centre is described. The formulae derived in the paper can be used to calculate the length of Airy's approximation of the orthodromic arc connecting two points on the sphere and on the Mercator chart. Moreover, the actual length of the orthodromic arc on the sphere and on the Mercator chart can be computed using the formulae derived in this paper. The purpose of the paper is not to suggest an application of Airy's method in navigation, but to analyse Airy's proposal and to show that a great circle arc on a Mercator chart is close to a circular arc for distances which are not too great. This property can be useful in education, having in mind that the stereographic projection is the only one that maps any circle on a sphere onto a circle in the projection plane.