Department of Geodesy and Geomatics Engineering, University of New Brunswick, 1973. – 153 p.The purpose of these notes is to give the theory and use of some common conformal map projections. The approach used is straightforward. It begins with the basics of differential geometry and conformal mapping. Then, given the conditions for a particular map projection, the particular conformal mapping equations are derived. This is a self-contained analytical approach.
The author is aware of attempts by at least one mathematician [Wray, 1973] to unify all classes of map projections such that the same set of mapping equations can be used to obtain all or many of the well-known projections, simply by assigning appropriate values to certain parameters in some generalized equations. This is somewhat of a simplified explanation of what actually happens, but it at least illustrates the generally of this contemporary approach. As of the date of writing these notes, the details (necessary for instruction at the undergraduate level) were not yet worked out and thus no use was made of it.
There is yet another attempt to obtain a generalized set of equations for conformal map projections alone. As explained in Section 9, this approach consists of one set of mapping equations which is capable of producing the well-known conformal projections: Mercator, Transverse Mercator, Lambert Conformal Conic, Stereographic and even other unnamed projections. This is achieved simply by assigning specific values for certain constants in a generalized set of equations. This approach is not readily usable since the expressions have not been developed to a sufficient degree of accuracy, and thus are not satisfactory for the practicing surveyor. Only a brief description of this approach is given in these notes.
More on the approach used herein. Complex arithmetic is exploited at every opportunity. Series expansions are avoided at occasions when the closed form exists. The reason being that computer centers nowadays have routines to evaluate natural logs, exponentiation, etc. Derivations are given to show the origin and important steps in the development of the main equations. Lengthy and detailed derivations are omitted from the text and reference made to an appropriate source or an appendix added.
These notes have been written under the assumption that the reader has knowledge of differential and integral calculus, complex arithmetic, ellipsoid geometry, and some knowledge of computer programming.
ContentsPreface
Table of contents
General
Introduction to conformal mapping
Review of complex variables
Review of differential geometry
Conformal projections in general
Mercator projection
Transverse Mercator projection
Lambert conformal conic projection
Stereographic projection
A generalized set of conformal mapping equations
Introduction to computations on a conformal map projection plane
Reduction of observations
Mathematical models for computation of positions
General formulae for reduction to the map projection plane
Specific formulae for reduction to various map projection planes
References
Appendices
