13 - The conformal model: Operational euclidean geometry

Abstract

The homogeneous model of Euclidean geometry is reasonably effective since it linearizes Euclidean transformations, and geometric algebra extends the classical homogeneous coordinate techniques nicely through its outermorphisms. In the next few chapters, the book presents the new conformal model for Euclidean geometry, which can represent Euclidean transformations as orthogonal transformations. Encoding those as versors gets the full power of geometric algebra, including the structure preservation of all constructions. This chapter starts the exposition by defining the new representational space (which has two extra dimensions and a degenerate metric) and showing what its vectors represent. The two extra dimensions are geometrically interpretable as the point at the origin (as in the homogeneous model) and the point at infinity (which nicely closes Euclidean geometry, making translations into rotations around infinity). It then focuses on how to represent the familiar flats and directions already present in the homogeneous model and how to move them around. As first applications, it uses the versor representation to provide straightforward closed-form interpolation of rigid body motions and universally valid constructions for the reflection of arbitrary elements. The natural coordinate-free specification of elements and operations in the conformal model is best appreciated using the interactive illustrations.