Projective Geometric Algebra
Projective Geometric Algebra
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
There is a separate
GitHub repository
that contains Mathematica packages for rigid, conformal, and spacetime geometric algebra.
Exercises for
PGA Illuminated
are currently being
posted on the wiki
in batches. Solutions to these exercises will be posted as well.
A small number of errata for the book are also
posted on the wiki
There is a
Discord server
for discussions about geometric algebra and PGA Illuminated in particular.
Projective Geometric Algebra Illuminated
Projective Geometric Algebra Illuminated
is a book for engineers, mathematicians, scientists, software developers, and anyone else
who wants to learn practical methods in geometric algebra. This book builds on new foundations and paints a more complete picture than any existing work.
It focuses on applications, it makes comparisons with conventional methods, and it candidly points out cases in which geometric algebra is not better.
Only a working knowledge of basic linear algebra is assumed.
The mathematical expressions in
Projective Geometric Algebra Illuminated
were remastered with the author’s
Radical Pie equation editor
in late 2025.
“A glorious book. Beautiful detail and impressive drawings.”
“Fantastic. No hand-waving to be found. I recommend Lengyel’s approach.”
“A thoroughly ergonomic read. Brings an important perspective: how to actually use projective geometric algebra in practical applications.”
Table of Contents
The 36×24 inch reference posters below contain a huge amount of information, including new research from 2020–2024.
Click on the images for PDF versions. High-quality prints of these posters are available in the U.S. at the links below each image.
Get print version
Get print version
This website is an
official silver sponsor
of the symbols ⟑ and ⟇, which have Unicode values U+27D1 and U+27C7.
These symbols denote the geometric product and geometric antiproduct in projective geometric algebra.
US