Projective geometric algebra (PGA) refers to a class of mathematical models in which geometric objects, operations for combining them in various ways, and

transformations that can be applied to them are all part of a single algebraic structure that is a specific type of Clifford algebra. This website deals with

two types of algebras called *rigid* geometric algebra and *conformal* geometric algebra, named for the types of transformations they contain.

Projective geometric algebra a natural fit for areas of computer science, especially computer graphics and robotics, that routinely use the mathematical concepts

that PGA encompasses.

The rigid geometric algebra (RGA) adds one dimension to ordinary space and incorporates representations for Euclidean points, lines, and planes as well as operations

for performing rotations, reflections, and translations. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates,

quaternions, and screw theory (which makes use of dual quaternions). The conformal geometric algebra (CGA) adds two dimensions to ordinary space and includes everything

in the rigid geometric algebra. It also adds representations for round objects like circles and spheres as well as dilations, inversions, and other conformal transformations.

Both types of algebra contain operations for easily joining lower-dimensional geometries to form higher-dimensional geometries, for calculating intersections among all of

the geometries, and for projecting one kind of geometry onto another.

This page is a central resource containing all of the work on the subject of geometric algebra by Dr. Eric Lengyel.

A large repository of mathematical reference materials can be found on the following two wikis:

• Rigid Geometric Algebra Wiki.

• Conformal Geometric Algebra Wiki.

A C++ library that implements much of this math is available under the MIT license on GitHub.

The 36×24 inch reference posters below contain a huge amount of information, including new research from 2020–2023. Click on the images for PDF versions.

High-quality prints of these posters are available on Amazon at the links below each image.

This website is an official silver sponsor of the symbols ⟑ and ⟇, which have Unicode values U+27D1 and U+27C7. |

New Developments in Projective Geometric Algebra These are the slides from the talk about PGA at the Game Developers Conference 2021. |