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Quaternions, Interpolation and Animation

Author: Erik B. Dam, Martin Koch and Martin Lillholm
Url: http://www.itu.dk/people/erikdam/DOWNLOAD/98-5.pdf
Format: Pdf
Year: 1998
Category: Mathematical Physics
Pages: 103
Clicks: 1577

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 fferent 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 different 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 fferential equations, whose analytical solution meets these constraints. Unfortunately, the set of diff erential equations cannot be solved analytically. As an alternative we propose a numerical solution for the differential equations. The diff 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.

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