A Swift Introduction to Projective Geometric Algebra

Video Description

This video is an introduction to Projective Geometric Algebra, which is a flavor of geometric algebra that allows for manipulating objects like points, lines, and planes, including operations like meets, joins, projections, and rigid transformations. This is done by treating lines/planes as their own linear space and doing geometric algebra on it. PGA provides a simple and powerful framework to do dimension-independent geometry. In case you were curious, this is finally the big project I've been teasing several times on my channel. It's also my #SoME3 submission. Here's my video on the linear space of lines: https://www.youtube.com/watch?v=T0b8-NJH79s In case you don't really know anything about geometric algebra, here are three videos of mine that I would suggest: https://www.youtube.com/watch?v=60z_hpEAtD8 (My original introduction) https://www.youtube.com/watch?v=0bOiy0HVMqA (The addendum to the introduction) https://www.youtube.com/watch?v=2AKt6adG_OI (The general definition of all of the operations used in this video) Here are some links to other introductions of PGA that might be useful/interesting: https://www.youtube.com/watch?v=ichOiuBoBoQ (The first resource that I saw that takes the "Rigid Transformations" approach to PGA) https://www.youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc (A playlist that covers PGA in a similar manner to the previous video but goes in more detail, and it culminates in describing rigid body dynamics) https://arxiv.org/abs/1101.4542 (One of the first sources to describe PGA, and it uses projective geometry) I would link to sources for the applications shown at the end, but you can find sources for most of them at bivector.net, so you can just go there instead. The notes got too big to fit in the description, so I moved them to a pinned comment. Discord: https://discord.gg/3Zj59zA2Rg Patreon: https://www.patreon.com/sudgylacmoe Patreon Supporters: Christoph Kovacs David Johnston Jason Killian LoganMP p11 Richard Penner Rosario trb Sections: 00:00 Introduction 02:15 The Linear Space of Lines 07:04 Basic Definition of 2D PGA 09:43 2D PGA Bivectors/2D Meets 13:47 2D PGA Points 14:57 More 2D Meets 16:15 2D Joins 17:50 2D Inner Product 19:35 2D Projections 21:51 2D Reflections 24:16 2D Rigid Transformations 26:27 2D Rigid Transformations on Points 28:03 2D Bivector Exponentials 31:16 2D Rigid Transformations Without PGA 32:54 2D Summary 33:27 3D Introduction 36:13 The Linear Space of Planes 37:32 Basic Definition of 3D PGA 38:07 3D PGA Bivectors and Trivectors 41:08 3D Meets 42:30 3D Joins 43:59 3D Inner Product 47:03 3D Projections 49:00 3D Rigid Transformations 49:46 3D Summary 50:25 nD PGA 50:54 Demonstration 52:10 Applications