PGA is a plane-based geometric algebra which appears eminently suited for the description and computation of operations in 3D Euclidean geometry. It naturally contains, in a real framework, efficient motion representations like homogeneous coordinates and dual quaternions, fully integrated with oriented object models. PGA's encoding of primitives and operators in blocks of four coordinates makes it very suitable for fast GPU implementation.

To satisfy the curiosity raised by the 2019 SIGGRAPH presentation on PGA, Leo Dorst decided to upgrade Chapter 11 of the 2007 book Geometric Algebra for Computer Science (GA4CS) to treat these new developments in a similarly precise but accessible style.



The present text assumes some basic geometric algebra knowledge (though it could be read without that, as a motivational inspiration to acquire that knowledge). Throughout, we try to be no more mathematical than required, and provide illustrations and links to software demonstrations.

This text takes care to propagate orientational signs and scaling factors through all its operations, since the author believes that PGA allows for such a consistently oriented Euclidean geometry. Guessing signs afterwards should no longer be required...

Enjoy! We plan to post updates on new developments on this page, and in upgraded versions of this 'Guided Tour to the Plane-Based Geometric Algebra PGA'.

Leo Dorst
Steven De Keninck

July 2020
March 2022 - We've updated our notation to better align with other tools, libraries and texts.