Projective Geometric Algebra

Projective geometric algebra is a four-dimensional mathematical model that naturally incorporates representations for Euclidean points, lines, and planes as well as operations for performing rotations, reflections, and translations in a single algebraic structure. It completely subsumes conventional models that include homogeneous coordinates, Plücker coordinates, quaternions, and screw theory.

I am Eric Lengyel, and this page is a central resource containing all of my work on the subject of geometric algebra. Geometric algebra is a specific type of Clifford algebra, and it includes the simpler Grassmann algebra.

A C++ math library that implements projective geometric algebra is coming soon. It will be available on this page.

The most recent developments in projective geometric algebra can be found in my blog:

Projective Geometric Algebra Done Right

Symmetries in Projective Geometric Algebra

The two 18×24 inch reference posters below contain a huge amount of information, including new research from 2020. They can be downloaded as a single PDF.

Projective Geometric Algebra, Part 1 Projective Geometric Algebra, Part 2

Foundations of Game Engine Development, Volume 1: Mathematics

This book, written in 2016, contains an entire chapter about projective geometry in Grassmann algebra. It is a detailed introduction to the subject that is the perfect starting place for anyone who wants to learn details about the wedge and antiwedge products.

Foundations of Game Engine Development website

Foundations of Game Engine Development, Volume 1: Mathematics
Grassmann Algebra in Game Development

Grassmann Algebra in Game Development

These are the slides from the talks about Grassmann algebra that I gave at the Game Developers Conference in 2012 and 2014.

Download slides as PDF

A Bigger Mathematical Picture for Computer Graphics

These are the slides from my keynote talk about Grassmann algebra at the WSCG conference in 2012.

Download slides as PDF

Slides and audio recording on YouTube

A Bigger Mathematical Picture for Computer Graphics