Appendix CConformal Mapping

The conformal property means that an angle between lines on the original equals the angle of their images. One must keep in mind that an angle is defined as the angle between tangents.

The first section begins with conformal mapping of planes using complex functions. It serves two purposes. First, it demonstrates in a rather simple manner the difference between conformality and similarity transformation. Second, it gives the technique for transforming the isometric plane into one of the desired standard conformal mappings, such as the ones by Mercator or Lambert. The next section gives the general formulation of conformality between general surfaces, making use of the first fundamental coefficients. Section C.3 gives the details about the isometric plane, and Section C.4 deals with those conformal mappings that are generally used in surveying. The most important ones are the transverse Mercator mapping and the Lambert conformal mapping. For example, all but one of the U.S. state plane coordinate systems are based on these mappings. An exception is a system in Alaska that uses the oblique Mercator mapping. The latter is not discussed here.

Clearly, conformal mapping has a long history with many individuals having made significant contributions. The historically inclined reader may consult the specialized literature for a full exposition of this interesting aspect. It might not be easy to delineate individual contributions in all cases. This is in part ...

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