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.

Keywords:

• Conformal transformations
• Helmert transformations
• Reverse transformations
• Inverse transformations

Acknowledgements

The author thanks his Director of Studies, Dr Brian Whiting, for his comments and suggestions. The author is grateful to the two anonymous reviewers for their comments and suggestions.

Notes on contributor

Andrew Ruffhead is a mathematician with master's degrees in Computing (Essex, 1977) and Geodesy (Oxford, 1985). Until retirement in 2005, he was a Senior Scientific Officer at the Defence Geographic Centre, U.K. As of January 2020, he is a PhD candidate researching geodetic datum transformations at the University of East London.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.