The theory of multivectors and the geometric product.
Curated by: All Angles (17 videos)
#geometricalgebra #projectivegeometry #cliffordalgebra Watch exclusive content on Patreon: https://www.patreon.com/user?u=86649007 So far, we have been looking at VGA (Vector Geometric Algebra), where basis vectors square to 1. But there are other options, such as PGA (Projective Geometric Algebra) which includes a basis vector that sqaures to zero. We will see how this allows us to model translations, and a point at infinity. We hint at many more possibilities. Geometric algebra is still a wide open domain that is actively researched. If you want to explore this vast frontier, we have a long list of resources that you can watch or download: [SDK 1] https://www.youtube.com/watch?v=ichOiuBoBoQ The best video by Steven de Keninck that introduces PGA, projective geometry, the connection between symmetries and their invariant objects, etc. Simply brilliant. [SDK 2] https://www.youtube.com/watch?v=tX4H_ctggYo Another very strong video by Steven de Keninck that gives you a good introduction to projective geometry and PGA. [SDK 3] https://www.youtube.com/watch?v=LQyKb0Flm3w Yet another amazing video by Steven de Keninck that introduces the homogenous coordinates of projective geometry, and then goes on to show how this leads to the concepts of orientation, and the unification of forces and torques in classical mechanics. You can also download the PDF at https://bivector.net/PGAdyn.pdf . [SUDGY 1] https://www.youtube.com/watch?v=0i3ocLhbxJ4 Sudgylacmoe: A swift introduction to projective geometric algebra. [SUDGY 2] https://www.youtube.com/watch?v=2AKt6adG_OI An overview of many different operations you can define on the elements of geometric algebra. [LIM 1] https://www.youtube.com/watch?v=PGZNYGwsXTw https://www.youtube.com/watch?v=ISKJPmuZkbY Good introductions to PGA and the meet & join operations. [AMD 1] http://www.faculty.luther.edu/~macdonal/GA&GC.pdf A Survey of Geometric Algebra and Geometric Calculus, by Alan MacDonald. [AMD 2] https://www.youtube.com/@AlanMacdonald1/playlists Alan MacDonald's Youtube channel, with videos about geometric algebra and geometric calculus. [BIV 1] https://www.youtube.com/@bivector https://bivector.net/ The Bivector channel on Youtube, and their web page with countless resources including implementations of PGA for programmers. You can support us on Patreon, where you can get access to exclusive content. Unfortunately, we won't be publishing any new videos after Geometric Algebra, at least not in the near future. Your support is still more than welcome of course: https://www.patreon.com/user?u=86649007 0:00 Introduction 1:33 Vectors that square to 1: reflections 2:12 Involutions 2:45 Vectors that square to -1: rotations 3:18 Vectors that square to 0: translations 4:48 Geometric numbers 5:54 The meet & join operations 6:55 Projective Geometric Algebra (PGA) 11:07 Shapes are also operations at the same time This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )