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:

Spherical geometry:

Hyperbolic geometry:

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

[thundrhawk]

Great explanations. If you haven't heard of it, there's a intimately related field called Projective Geometric Algebra. It applies the constructions of Geometric Algebra to Projective Geometry. This allows you to easily define projective points as vectors, projective lines as bivectors, and projective planes as trivectors. This allows you to use the wedge product in GA similarly to how you were using the direct sum, which also makes the construction at 18:40 come a little easier. B is the bivector representing the plane spanned by the two vectors in A (this also follows from how B is the normal vector to the plane spanned by A, and normal vectors don't transform as vectors but rather as bivectors).
Eric Lengyel has done research in expanding PGA to include a new operation he called the antiwedge product, which performs an operation analogous to the intersection use here. Along with some other operations. It gets complicated extremely quickly, but it's a neat rabbit hole to follow. He's also applied PGA to video game math in his C4 Engine.

[FiniteJest]

Homogenization is a great tool to help students understand conic sections. I always learned about the projective plane more abstractly as a manifold. I’m really happy to see concrete computations. Thanks for creating this video.

[friese]

Really liked the video! Please continue!
One of the few videos going into the details of how to do things hands on.

[SchlafliedSensor]

This is one of the best and coolest videos that helped me understand the matter of my lectures for my studies better. Thanks, CoolComputery!

[maxpercer]

This is great, many advanced treatments of projective geometry rush through nitty gritty details (such as the ray versus the line issue for a point at infinity).

[TheJara]

Great video, please keep posting, sooner or later the channel will be top place for your topics...since you have all the necessary ingredients.

[theproofessayist]

Ah another math youtuber who knows of Wildberger - good to see! I really do not like Wildberger's anti real analysis rants because they don't make sense to me and he seems unpleasantly angry when doing them but his videos on rational trigonometry and projective geometry are such a gem. I am actually onboard with his revisionist project of trigonometry if the curricula were to ever change like that in the future - it's been awhile since I watched the videos but all I could remember was from a pedagogical perspective his arguments were solid in my book. He really has made great contributions to geometry and algebra.

[StarCommandTrainingModules]

This is so helpful man. Thank you for this.

[Number_Cruncher]

Thank you for your throughout introduction.

[OldSloGuy]

Very succinct. Not very good as a learning tool alone, but that's OK. Learning is a bit like throwing mud at a Teflon wall, very little sticks initially, repetition is necessary. Another problem is the viewpoint, seeing the forest while surrounded by trees. A typical classroom instruction is like nailing two sticks together with a hammer without ever visualizing the house that the skill is associated with. It's the difference between typical academic presentations and an apprenticeship. The classroom represents a tool kit, without demonstrations of appropriate uses, that supposedly comes after graduation with mentoring from professionals, the apprenticeship. So, thank you for providing this outline of the subject so the curious among us have an overview.

[gregoriuswillson]

Very cool animations, this help me too understand better in projective geom, may i know what referrence u use this since it seem it is pretty interesting from ur POV

[robharwood]

Hey. Most of the links in your video description have been corrupted -- probably from copy-pasting them from their abbreviated form (with the three-periods ellipses '...' trailing at the end of each abbreviated link, breaking it). When pasting text with links in it into YT, it works best if you keep an 'master'/original version (of, say, your video description) and always copy from that and paste into YT. Avoid copying from YT's abbreviated version and pasting back into YT.

Anyway, enjoyed the video. It's very very dense with info! I bet you could easily produce several more videos, each on just one or two particular aspects of the topic, fleshed out a little, and you could have plenty of content to post on your channel for a while! 🤓

Wildberger does indeed develop a similar system using projective geometry to support spherical and hyperbolic geometry. You might be interested to see his stuff on what he calls 'chromogeometry', which in a sense unifies the Euclidean and hyperbolic metrical notions into a common framework.

Also, he applies it to his 'rational trigonometry', to the extent that he provides essentially the identical proof of (for example) Pythagoras' Theorem for Euclidean, hyperbolic, and essentially any other quadratic-form metric you want to cook up. Pretty cool.

[cmilkau]

In finite dimensions, clmspace(A) = nullspace(B*) iff clmspace(A) is the orthogonal complement of clmspace(B). Here B* is the adjoint of B, if your inner product is the dot product that's just the transpose. Basically, nullspace(B*) are literally all vectors zeroing the inner product with columns of B, i.e. its orthogonal complement.

[NonTwinBrothers]

I had no idea webdriver torso was into geometry. Amazing!

[ibrahimmdumariqbal]

It really helps me
Lots of love from bengal

[techconbd]

Conics by Keith Kendig
Is where people should go after watching this awesome video

[tanchienhao]

You should submit this to SoME2! Awesome exposition on projective geometry

[paxdriver]

The projected line at infinity is precisely offset by the height of the eye and never sees the ground. It's infinite because it doesn't end looking at the ground. Looking up would ja e the same effect as looking parallel.

Instead of considering the infinite boundary as seeing the ground projection, it's more accurate to relate it to the boundary line from which it is said that you are no longer looking at the ground than it is to say you're looking parallel to the ground. It's too high to see down so it never sees the road. It doesn't include the road, ergo this line is not included in the views of the road.

I'm not sure this analogy carries tbh, maybe I'm missing something.

[melissapereira]

can you please do a video about affine geometry ?

[stolfjr]

It is very confusing when you use "/", "*" and ":" randomly as a notation, why did you write the equations as a text like "(a(x/z)+b(y/z)+c)*z=0*z instead of in a proper equation-like, visual way?