New Material

On this page we publish new material that would be included in the second edition of the book.

PGA. July 6, 2020

In Chapter 11, we treated the GA version of Homogeneous Coordinates to encode Euclidean geometry. We found that lacking in some aspects--we did not have the rotors for motions, nor a convenient computation dual--and so we moved on to CGA, the conformal model, of two more dimensions than the Euclidean base space.

Since then, we have learned that Euclidean rotors can be constructed in a space of only one more dimension, in an algebra that represents Euclidean planes as vectors: PGA. Here is a Guided Tour, which can be viewed as a (rather extensive!) replacement for our old Chapter 11.

New blade factorization algorithm, new Join algorithm. 2008

We have found a new way to factor blades that also helps to speed up computing the join of blades. Computing the join of blades is now 10x faster than the algorithms presented in the book (as benchmarked in Section 5.11.2).

A rewritten version of this paper would replace Sections 21.6 and 21.7 in a second edition of the book. The paper is rather dense because of the 10-page limit set by the AGACSE 2008 conference, so be sure to read those sections first!

Extended Treatment of Meet and Join. 2007

One of our readers, Greg Grunberg, found our treatment of the Meet and Join operators in Chapter 5 interesting, but terse. He has written a more extended discussion of some of the basic properties, including explicit proofs and derivations. We are grateful that he has allowed us to post this on our book website.