9 - Modeling geometries

Abstract

More complicated geometrical objects do not require new operations or techniques in geometric algebra, merely the standard computations in a higher-dimensional space, followed by an interpretation step. The geometric algebra approach considerably extends the classical techniques of homogeneous coordinates, so it pays to redevelop this fairly well-known material. The better model to treat the metric aspects of Euclidean geometry is a representation that can make full use of the power of geometric algebra. That is the conformal model which requires two extra dimensions. It provides an isometric model of Euclidean geometry. In this representation, all Euclidean transformations become representable as versors, and are therefore manifestly structure-preserving. This gives a satisfyingly transparent structure for dealing with objects and operators, far transcending the classical homogeneous coordinate techniques. Initially, it shows how this indeed extends the homogeneous model with metric capabilities, such as the smooth interpolation of rigid body motions. Then it is found that there are other elements of Euclidean geometry naturally represented as blades in this model: spheres, circles, point pairs, and tangents. These begin to suggest applications and algorithms that transcend the usual methods. In the last chapter on the conformal model, the reason behind its name is found: all conformal (angle-preserving) transformations are versors, and this now also gives the possibility to smoothly interpolate rigid body motions with scaling. In all of these chapters, the use of the interactive software is important to convey how natural and intuitive these new tools can become.