Equivalence properties of 3D conformal transformations and their application to reverse transformations

Abstract

Seven-parameter conformal coordinate transformations, also known as Helmert transformations, can be constructed in more than one way. Two possible orderings of the rotations are in common use, giving rise to Helmert versions 1 and 2. It is demonstrated how the rotation parameters of either version can be converted into the rotation parameters of the other. This is useful when software is designed for the other version. It also enables computation of the same-formula inverse transformation by changing the sign of the equivalent ‘other version’ parameters. These results were primarily intended for conformal transformations between geodetic datums. They can, however, be extended to coordinate transformations in disciplines such as photogrammetry where rotations sometimes exceed 90 degrees.

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