Different approaches/notation for Projective Geometric Algebra from Eric Lengyel and Jon Selig

Google "Projective Geometric Algebra" and you see that there are two different notations people are using. This video tries to clarify things around that!

Predictably, I made a few mistakes! I did a stream clarifying these but the connection let me down. So, here are those mistakes:

1. I say in this video that you can measure the angle between anything and anything, without dualization. That's not true! There is an interesting exception, which is that you can't measure the angle between points or lines at infinity.

2. I keep saying "polar space" here, but that's muddling the terminology. Ordinary space is "euclidean space", the other space (the tetris window full of lines) is actually called "dual euclidean", and then the term "polar" is meant to be used for the two of them added together. So "polar tetris" is the right name for the game, since it has both, but the window full of lines alone is "dual euclidean tetris", not "polar tetris"

3. I didn't use the word "antispace" here at all, which is an oversight. So I'll say it here: "anti space", which is Eric's term, is another term for the tetris window, whereas the tetris window full of lines is what he calls "space". Of the things that start with "anti-" in PGA (anti geometric product, anti reverse), all of them are specific to Eric's approach, with two exceptions: the antiwedge, which others call the join or regressive product, and the antinorm, which others call the ideal norm. -- Watch live at   / hamish_todd