GA2015 – Lecture 7

Conformal Geometric Algebra

It is only fairly recently that we realised just how powerful the combination of conformal geometry and geometric algebra could be as a tool for understanding Euclidean geometry. The key ideas were introduced around 2000, and have led to a resurgence in geometric algebra.

In this lecture I have attempted to motivate the key idea behind conformal geometric algebra from a novel starting point. It is common to use the stereographic projection as a starting point, but instead we start with the idea that we want to represent points as vectors where the inner product is proportional to the square of the distance between points. This naturally leads to the idea of points as null vectors.

From here we introduce the point at infinity and pick out an origin, recovering the generators of Euclidean geometric algebra in the process. Lines are represented as trivectors, and turn out to be special cases of circles that pass through infinity. Sphere and planes are similarly handled in a unified manner.

Rotors are constructed that generated translations and dilations, as well as rotations, and the addition of inversion shows how rotors generate the full conformal group of angle-preserving transformations. We end with some examples of intersection tests and reflection operations, demonstrating how complex geometrical relationships can be summarised in simple algebraic expressions.