Last updated: 2026-09-26

M
Masters level
EPH
Ephemeral / Tooling — Knowledge that evolves in months to a year — check for updates

Appendix: Further Reading and Live Demos

A.1 How to Use This List

The chapters of this series build one line of argument. The sources below reach the same ground from other directions, and several let you move things on screen instead of reading about them. They're grouped by what you probably want next: a gentler way in, the projective model of Chapter 11, the conformal model of Chapter 8, live demos, and work beyond graphics.

A path that works for a programmer new to the subject: the ten Bosch article for the picture, then Chapter 9 and Chapter 11 for the algebra, then the 3D PGA cheat sheet and the Dorst and De Keninck tour for the full toolkit, and the dynamics paper when you want to simulate something.

A.2 A Gentler Way In

  • Let's remove Quaternions from every 3D Engine, by Marc ten Bosch. An interactive introduction to rotors, with diagrams and video. It builds up bivectors and the geometric product, then shows two reflections composing into a rotation, the same construction Chapter 11 uses for motors. Best first stop if you want to see the geometry move.
  • Less Weird Quaternions, by Malte Skarupke (5 August 2017). Reads a quaternion as a scalar plus three bivectors, which is the view Chapter 9 reaches by proof, and explains why the double cover helps interpolation.
  • Geometric Algebra Primer, by Jaap Suter (2003). A short introduction built from two ideas, the multivector and its geometric product. The copyright notice allows personal and classroom copies, so it's linked here rather than reproduced.
  • Imaginary numbers are not real: the geometric algebra of spacetime, by Gull, Lasenby and Doran, Foundations of Physics 23(9), 1993. A tutorial from the physics side. It shows how the geometric product of vectors in two and three dimensions gives the imaginary numbers of conventional methods a geometric meaning, and it introduces spacetime algebra.

A.3 Projective GA: Notes, Cheat Sheets, and Dynamics

  • Geometric Algebra for Computer Graphics, the SIGGRAPH 2019 course notes by Charles Gunn and Steven De Keninck (also the ACM course entry). The standard graphics-oriented introduction to PGA.
  • The 2D PGA and 3D PGA cheat sheets, by Gunn and De Keninck, one page each. They tabulate the basis and multiplication table, the incidence, projection and metric formulas, and motors, with a rigid-body mechanics panel. The 3D sheet writes a point as \(x\,\mathbf{e}_{032} + y\,\mathbf{e}_{013} + z\,\mathbf{e}_{021} + \mathbf{e}_{123}\), the same element as the point in Chapter 11 under a different blade ordering, and it uses the same plane-based convention: the outer product meets, the regressive product joins.
  • A Guided Tour to the Plane-Based Geometric Algebra PGA, by Leo Dorst and Steven De Keninck (version 2.0, March 2022). About 100 pages covering 2D and 3D PGA with an eye on implementation, and begun as a replacement for Chapter 11 of the 2007 book below. It assumes some familiarity with standard geometric algebra.
  • "May the Forque Be with You": n-Dimensional Rigid Body Dynamics in PGA, by Dorst and De Keninck. Applies PGA to Newton's laws so that linear and angular quantities are treated together, in any number of dimensions. It's the natural next step after Chapter 11 if the goal is physical simulation, and it comes with ganja.js implementation examples.

A.4 Conformal GA and the Reference Book

  • A Covariant Approach to Geometry using Geometric Algebra, by Anthony Lasenby, Joan Lasenby and Richard Wareham (Cambridge University Engineering Department technical report CUED/F-INFENG/TR-483). Develops conformal GA as a five-dimensional representation of 3D space in which spheres, circles, lines and planes are elements, with examples from computer graphics and a discussion of its advantages and disadvantages as a programming language. It's the natural companion to Chapter 8.
  • Geometric Algebra for Computer Science, by Dorst, Fontijne and Mann (Morgan Kaufmann, 2007), and its companion site. The site offers interactive versions of nearly all the book's figures, downloadable source code for the programming exercises, solutions to selected exercises, and errata.

A.5 Live Demos: The ganja.js Playground

The ganja.js example playground is a menu of runnable examples you can edit in the browser. It covers more algebras than the ones in this series. The names below are the playground's own, and each link opens that example directly.

Chapter 11's own worked examples run in the same library, in section 11.11.

A.6 Beyond Graphics

Geometric algebra reaches well past rendering. The 2026 theme issue of Philosophical Transactions of the Royal Society A (volume 384, issue 2326) collects recent work on modern applications. Its articles include an overview of implementation representations and their memory requirements, null geometric algebra, geometric algebra for quadrics, integral transforms, coincidence site lattices, applications in cryptography and machine learning, and power-systems engineering. The implementation overview bears directly on the trade-offs of Chapter 11's section on libraries.