3D Coordinate Transformation by using Quaternion Algebra

Abstract

Rotation in space is one of the most important problems in mathematics and other sciences. Respectively, the three-dimensional coordinate transformations 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.

After analyzing the mathematical context of point rotation in space, this chapter presents an investigation of specific data using three different transformation methods. The method of Euler angles, quaternion, and dual-quaternion algebra is used. After research, three artificial sets of data, which were structured in a specific way and forced into specific transformations, were used to find out the sensitivity of each method. In addition, three real transformation problems, concerning monitoring and deformation, were tested, to have an accurate result of which method is best. The problems  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.