An Intuitive Introduction to Projective Geometry Using Linear Algebra

This is an area of math that I've wanted to talk about for a long time, especially since I have found how projective geometry can be used to formulate Euclidean, spherical, and hyperbolic geometries, and a possible (and hopefully plausible) way projective geometry (specifically the model that uses lines, planes, etc. through the origin) could have been discovered and not just created out of thin air.

I am most likely not the first person to discover what I say in this video, but I have not found any sources that explicitly state the same things (except possibly NJ Wildberger with his video on how hyperbolic geometry is "projective relativistic geometry", which I haven't watched, but judging from the thumbnail it seems like he found the same connection between projective geometry and the Minkowski model of hyperbolic geometry that I make in this video).

The first half of this video is intended for everyone; the second half (where I start talking about linear algebra) is intended for those who already know that subject on an introductory level, e.g. those who have taken a class in it or have watched 3Blue1Brown's series on it.

Everything in this video comes from bits and pieces of articles and videos that I have sporadically watched over the last several (maybe 6 or 7) years, plus linear algebra that I have learned in a class I took more recently. As a result, I probably cannot give a complete list of all the sources I have used, but I will list as many as I can remember down below:

Projective geometry:
https://en.wikipedia.org/wiki/Homogen...
   • Projective geometry and homogeneous coordi...   ("Projective geometry and homogeneous coordinates | WildTrig: Intro to Rational Trigonometry", Insights into Mathematics)

Spherical geometry:
https://en.wikipedia.org/wiki/Spheric...
https://brilliant.org/wiki/spherical-...

Hyperbolic geometry:
https://en.wikipedia.org/wiki/Hyperbo...
   • Projection from Hyperboloid to the Beltram...   ("Projection from Hyperboloid to the Beltrami–Klein disk.", Jamnitzer)
https://dl.tufts.edu/concern/pdfs/bk1... ("Hyperbolic Geometry on a Hyperboloid", William F. Reynolds)
https://www.roguetemple.com/z/hyper/m... ("Models and projections of hyperbolic geometry", Rogue Temple)

2D and 3D plots were made with Desmos and GeoGebra, respectively. All other images were made by me in Google Slides.

Chapters:
PART 1
0:00 Intro
0:31 Defining projective points and lines
4:19 Spatial coordinates
7:11 Projective quadratics
8:40 Non-Euclidean geometries
10:52 Distance metrics

12:11 PART 2 (linear algebra)
12:33 Defining projective points, lines with linear algebra
13:47 clmspace vs. nullspace representation of projective linear objects (points, lines, planes, ...)
16:32 clmspace to nullspace representation of a projective line (includes cross product)
20:31 Spans of clmspaces and interseections of nullspaces
21:33 3D projective geometry
23:13 Projective quadratics and double-cones

26:34 Summary

#SoME2