Geometric algebra and its application to computer graphics

Abstract

Early in the development of computer graphics it was realized that projective geometry is suited quite well to represent points and transformations. Now, maybe another change of paradigm is lying ahead of us based on Geometric Algebra. If you already use quaternions or Lie algebra in additon to the well-known vector algebra, then you may already be familiar with some of the algebraic ideas that will be presented in this tutorial. In fact, quaternions can be represented by Geometric Algebra, next to a number of other algebras like complex numbers, dual-quaternions, Grassmann algebra and Grassmann-Cayley algebra. In this half day tutorial we will emphasize that Geometric Algebra • is a unified language for a lot of mathematical systems used in Computer Graphics, • can be used in an easy and geometrically intuitive way in Computer Graphics. We will focus on the (5D) Conformal Geometric Algebra. It is an extension of the 4D projective geometric algebra. For example, spheres and circles are simply represented by algebraic objects. To represent a circle you only have to intersect two spheres ( or a sphere and a plane ), which can be done with a basic algebraic operation. Alternatively you can simply combine three points (using another product in the algebra) to obtain the circle through these three points. Next to the construction of algebraic entities, kinematics can also be expressed in Geometric Algebra. For example, the inverse kinematics of a robot can be computed in an easy way. The geometrically intuitive operations of Geometric Algebra make it easy to compute the joint angles of a robot which need to be set in order for the robot to reach its goal.