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/Homogeneous_coordinates
https://www.youtube.com/watch?v=q3turHmOWq4 ("Projective geometry and homogeneous coordinates | WildTrig: Intro to Rational Trigonometry", Insights into Mathematics)
Spherical geometry:
https://en.wikipedia.org/wiki/Spherical_geometry
https://brilliant.org/wiki/spherical-geometry/
Hyperbolic geometry:
https://en.wikipedia.org/wiki/Hyperboloid_model
https://www.youtube.com/watch?v=KO5eE5d59vQ ("Projection from Hyperboloid to the Beltrami–Klein disk.", Jamnitzer)
https://dl.tufts.edu/concern/pdfs/bk128p14r ("Hyperbolic Geometry on a Hyperboloid", William F. Reynolds)
https://www.roguetemple.com/z/hyper/models.php ("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