Geodetic applications of conformal transformations in three dimensions

Abstract

Conformal transformations in two dimensions have played a leading part in the solution of potential problems, and in the theory of map projections. They have hardly been used at all in three dimensions, no doubt because possible transformations between two flat 3-spaces are mostly trivial. Much greater freedom in the choice of scale factor is however obtained if, as a mathematical device, one space is not restricted to being flat. The differential geometry of the transformation is worked out on this basis.

The main problems in modern theoretical geodesy fall into the following three categories and their interactions:

• (a) A simple and accurate description of the Earth’s gravitational field, particularly the lines of force (or vertical lines) and the equipotential (or level) surfaces, and its application to the reduction of geodetic measurements.

• (b) The dynamics of near-Earth satellites, in view of their growing use for angular and distance measures and for exploring the gravitational field.

• (c) The propagation of light or electro-magnetic waves in a refracting medium, in which all surface measures of length and direction are made.

It is shown that all three problems are mathematically equivalent to the study of geodesics and geodesic parallels in curved conformal 3-space; so that any advance made in one direction, and indeed the corpus of existing knowledge in each subject, can immediately be applied to the others. More details will be supplied in later papers.