Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra

Abstract

The three-dimensional coordinate’s transformation from one system to another, and more specifically, the Helmert transformation problem, is one of the most well-known transformations in the field of engineering. In this paper, its solution, in reverse problem, was investigated for specific data using three different methods. It is presented by solving it with the method of Euler angles as well as with the use of quaternion and dual-quaternion algebra, after first giving some basic mathematical theory. After research, not only were three artificial sets of data used, which were structured in a specific way and forced into specific transformations to be solved, but also a real geodesy problem was tested, in order to identify the sensitivity and problems of each method. Statistical analysis of the results was performed by each method, while it was found that there were significant deviations in rotations and translations in the method of Euler angles and dual quaternions, respectively.