Quaternions, Interpolation and Animation

The main topics of this technical report are quaternions, their mathematical properties, and how they can be used to rotate objects. We introduce quaternion mathematics and discuss why quaternions are a better choice for implementing rotation than the well-known matrix implementations. We then treat di erent methods for interpolation between series of rotations. During this treatment we give complete proofs for the correctness of the important interpolation methods Slerp and Squad . Inspired by our treatment of the di erent interpolation methods we develop our own interpolation method called Spring based on a set of objective constraints for an optimal interpolation curve. This results in a set of di erential equations, whose analytical solution meets these constraints. Unfortunately, the set of di erential equations cannot be solved analytically. As an alternative we propose a numerical solution for the di erential equations. The di erent interpolation methods are visualized and commented. Finally we provide a thorough comparison of the two most convincing methods (Spring and Squad). Thereby, this report provides a comprehensive treatment of quaternions, rotation with quaternions, and interpolation curves for series of rotations.