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Quaternions, Interpolation and Animation

This has to do with receiving proper credit for the work and being able to add it to your design portfolio. You should ask for a credit line to be included in the work itself. You should state that, once the project has been completed and introduced to the public, you will have the right to add the client’s name to your client list and the right to enter the work into design competitions. You’ll also want to be able to show and explain portions of the completed project to other companies when you are pitching new business. Sometimes clients. | Quaternions Interpolation and Animation Erik B. Dam erikdam@diku.dk Martin Koch myth@diku.dk Martin Lillholm grumse@diku.dk Technical Report DIKU-TR-98 5 Department of Computer Science University of Copenhagen Universitetsparken 1 DK-2100 Kbh 0 Denmark July 17 1998 Abstract 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 betw een 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 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. i Contents 1 Introduction 1 2 Geometric transformations 3 2.1 Translation. 3 2.2 Rotation. 3 3 Two rotational modalities 5 3.1 Euler angles. 5 3.2 Rotation matrices. 6 3.3 Quaternions. 7 3.3.1 Historical background. 7 3.3.2 Basic quaternion mathematics . 8 3.3.3 The algebraic properties of quaternions. 12 3.3.4 Unit quaternions. 14 3.3.5 The exponential and logarithm functions . 15 3.3.6 Rotation with quaternions. 17 3.3.7 Geometric intuition . 22 3.3.8 Quaternions and