Projective geometric algebra

A modern framework for doing geometry

Welcome to the resource site for projective geometric algebra. Please contact me (gunn at if you have questions or suggestions.


Projective geometric algebra is a version of geometric algebra built on top of projective space. It provides a direct implementation of the Cayley-Klein construction of metric spaces within projective space. The resulting framework is powerful, compact, expressive, light-weight, and practical to implement. The above figures illustrate some of the richness of the geometric product for the case of the euclidean plane (left) and the hyperbolic plane (right), while the middle figure depicts the differentiated structure in dimension 3 of the isometry group (independent of metric) and its important components.


Almost all the necessary ingredients for PGA were already known to Clifford and Klein in the 19th century, but technical difficulties in the euclidean case postponed its discovery. It was introduced into the modern literatur by Jon Selig in 2000 and has been developed further by Charles Gunn in his Ph. D. (Technical University Berlin, 2011) and a series of articles listed below.


Most of the resources provided below focus on the euclidean case, as it is the case of most practical interest. Here are the relevant links:

Slides for the tutorial Projective geometric algebra: A Swiss army knife for graphics and games presented at Geometry Summit 2016, Free University Berlin, June 19, 2016.

"Doing euclidean plane geometry using projective geometric algebra" is the author's copy of recently published article that introduces PGA in the context of euclidean plane geometry.

"Geometric algebras for euclidean geometry" is the author's copy of a recently published article that compares different geometric algebras for doing euclidean geometry.

My Ph. D. thesis "Geometry, Kinematics, and Rigid Body Mechanics in Cayley-Klein Geometries" that describes the full gamut of projective geometric algebra, both euclidean and non-euclidean, culminating in a metric-neutral treatment of rigid body mechanics.

An article containing the Euclidean content of my thesis , presented at AGACSE 2010 Amsterdam and published as Ch. 15 of "A Guide to Geometric Algebra in Practice", Springer, 2011, edited by Dorst and Lasenby.

An extended version of the previous article as ArXiV preprint.

"Rational trigonometry via projective geometric algebra", a short article reporting on my successful efforts to express Normal Wildberger's rational trigonometry in the language of PGA.

Blog post showing how to use projective geometric algebra to solve a geometric exercise in the euclidean plane. This construction is featured in the article on doing euclidean plane geometry (above).

There is an excellent Javascript implementation of PGA (and other members of the GA family) which comes with a full set of online examples. Thanks to Steven de Keninck for this important contribution.