Dam E.B., Koch M., Lillholm M. Quaternions, Interpolation and Animation

Copenhagen: University of Copenhagen, 1998. — 103 p.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 different methods for interpolation between series of rotations. During this treatlnent we give complete proofs for the correctness of the important interpolation methods Slerp and Squad. Inspired by our treatmnent 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 differential equations, whose analytical solution meets these constraints. Unfortunately, the set of differential equations cannot be solved analytically. As an alternative we propose a numerical solution for the differential equations. The different 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.