Source
- URL: https://x.com/EricLengyel/status/1209615013725396992
- Author: Eric Lengyel (@EricLengyel)
- Posted: 2019-12-24 23:21:07
Thread
1/ @EricLengyel
1/8 Every operation in Grassmann/geometric algebra has a dual, which is just as important. The dual of the wedge “progressive” product is the antiwedge “regressive” product.
2/ @EricLengyel
2/8 With few exceptions, texts on geometric algebra completely ignore the antiwedge product and often have to compensate with less fundamental operations like the unintuitive “contraction” product. I get the impression that many authors don’t actually understand the connection.
3/ @EricLengyel
3/8 There is also a dual to the dot product and a dual to the geometric product. Nobody seems to know anything at all about those, and you won’t find them in any text. However, they are important for properly understanding operators that perform rigid motions.
4/ @EricLengyel
4/8 Because nobody knows the dual geometric product even exists, we are currently using dual quaternions incorrectly. I’m not saying the actual rotation/translation calculations end up being wrong.
5/ @EricLengyel
5/8 But the wrong types of mathematical entities have to be jammed into the calculations to compensate for the fact that the wrong fundamental operations are being used.
6/ @EricLengyel
6/8 This is something that has always bothered me about dual quaternions. After some recent research, I’m confident that I now understand what’s going on, and I have successfully reformulated everything as necessary.
7/ @EricLengyel
7/8 I don’t think it’s that hard to correct the situation, but it does require that you recognize the dual to the geometric product. When I’ve tried to talk to the GA community about this in the past, however, they cover their ears and yell “La, la, la, la…”.
8/ @EricLengyel
8/8 I’ll write something up about this soon. It will also be covered in FGED3.
Branches: 2019-12-25-gfxcoder-precisely-the-reason-i-bought-fged1-looking, 2019-12-25-sterlingcrispin-when-you-say-ga-community-do-you-mean-that, 2019-12-25-Atridas87-my-problem-with-the-anti-wedge-product-is-that-it, 2019-12-25-vicen_cm-yes-please