sudgylacmoe | A Swift Introduction to Projective Geometric Algebra @sudgylacmoe | Uploaded August 2023 | Updated October 2024, 4 hours ago.
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: 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:
youtube.com/watch?v=60z_hpEAtD8 (My original introduction)
youtube.com/watch?v=0bOiy0HVMqA (The addendum to the introduction)
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:
youtube.com/watch?v=ichOiuBoBoQ (The first resource that I saw that takes the "Rigid Transformations" approach to PGA)
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)
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: discord.gg/3Zj59zA2Rg
Patreon: 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
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: 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:
youtube.com/watch?v=60z_hpEAtD8 (My original introduction)
youtube.com/watch?v=0bOiy0HVMqA (The addendum to the introduction)
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:
youtube.com/watch?v=ichOiuBoBoQ (The first resource that I saw that takes the "Rigid Transformations" approach to PGA)
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)
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: discord.gg/3Zj59zA2Rg
Patreon: 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
