Complex Numbers and Quaternions – A unified view on representations and metrics for rotations

Bertold Bongardt

DFKI GmbH DFKI Documents (D) 14-03 14-03 ISBN ISSN 0946-0098 Selbstverlag, Bremen 6/2014.


On this poster, a novel view on complex numbers and quaternions is motivated by introducing a five dimensional complex space which is defined as the union of the complex plane C with the quaternion space H. For rotations with a fixed rotation axis, the complex 5-space can be visualized by R3 and by R2. In these visualizations, the representations of a rotation via a complex number, quaternions, and a rotation matrix appear in an elementary-geometric setup generalizing the unit circle. The definition of the complex 5-space is based on an explicit distinction of four different imaginary units. The poster illustrates one usage of these novel concepts with a comparison of distance measures for rotational displacements.

German Research Center for Artificial Intelligence
Deutsches Forschungszentrum für Künstliche Intelligenz