Eurographics 2004. Tutorial 3 - Geometric Algebra and its Application to Computer Graphics

In this tutorial we will give an overview of Geometric Algebra and its application to computer graphics. First of all, we want to motivate the topic and give insights into some applications. In particular, the Conformal Geometric Algebra with its so-called 'conformal model' of 3-dimensional Euclidean geometry will be introduced. In this model, Euclidean objects and their interactions will be explored and visualized interactively. With help of the conformal model we will describe animations and motions. It will be shown how it can be used quite advantageously to treat this kind of computer graphics applications. We will give some basic visual examples and describe rigid body motions and their interpolations. We will focus on the inverse kinematics and dynamics of kinematic chains in order to describe motions of robots and human figures. At the university of Amsterdam a ray tracer was developed in order to compare different geometric approaches from the implementation and performance point of view. Compared to linear algebra, the richer mathematical language of GA leads to more work for implementing the algebra, but less work for implementing the application. We discuss the issues in implementing a numerical geometric algebra package for a language like C++. We compare various existing implementations and look at their performance. We conclude with future implementation methods like SIMD hardware suitable for GA and generative programming. During the tutorial only the most fundamental mathematical aspects of Geometric Algebra will be presented. This is possible, since most aspects of Geometric Algebra can be understood with geometric intuition. The actual mathematical 'inner workings' of the algebra will be detailed in an accompanying script that also contains many visual examples from the presentations. The tutorial will be rounded off by an outlook into possible future applications of Geometric Algebra in computer graphics.