An Intuitive Introduction to Projective Geometry Using Linear Algebra
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- Опубліковано 18 чер 2024
- 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:
en.wikipedia.org/wiki/Homogen...
• Projective geometry an... ("Projective geometry and homogeneous coordinates | WildTrig: Intro to Rational Trigonometry", Insights into Mathematics)
Spherical geometry:
en.wikipedia.org/wiki/Spheric...
brilliant.org/wiki/spherical-...
Hyperbolic geometry:
en.wikipedia.org/wiki/Hyperbo...
• Projection from Hyperb... ("Projection from Hyperboloid to the Beltrami-Klein disk.", Jamnitzer)
dl.tufts.edu/concern/pdfs/bk1... ("Hyperbolic Geometry on a Hyperboloid", William F. Reynolds)
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
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.
sudgylacmoe just released a very good introductory video on Projective Geometric Algebra :) ua-cam.com/video/0i3ocLhbxJ4/v-deo.html
> the antiwedge product, which performs an operation analogous to the intersection use here
In regular PGA, the meet of two lines is just their wedge product. It's not clear for me what Lengyel's formulation brings to the picture.
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.
Really liked the video! Please continue!
One of the few videos going into the details of how to do things hands on.
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).
This is one of the best and coolest videos that helped me understand the matter of my lectures for my studies better. Thanks, CoolComputery!
Great video, please keep posting, sooner or later the channel will be top place for your topics...since you have all the necessary ingredients.
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.
This is so helpful man. Thank you for this.
Thank you for your throughout introduction.
Awesome vid!
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.
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.
Thanks for notifying me about the corrupted links: I indeed have a master version of the video description and have just copy-pasted it again onto UA-cam; the links still display '...' but they are working now (as of writing this comment).
I could produce several smaller videos, but I wanted to create a single self-contained video that covered the entire process of formulating projective geometry, so that anyone who watches this video could potentially use it for any possible applications of geometry (e.g. making a 3D engine). Plus, one video is (in my opinion) usually more convenient than several smaller videos (even in a playlist), and I'm not sure if #SoME2 allows a playlist of multiple videos to be an entry. Lastly, my next few planned videos (if I manage to make them) will be on completely different topics in math.
I'll be sure to look more into Wildberger's rational trigonometry and chromogeometry!
I had no idea webdriver torso was into geometry. Amazing!
Thanks! I notice that on your about page you have a link to a Desmos page showing perspective projection of a parabola. www.desmos.com/calculator/5kalh6a9b3
@@coolcomputery Yup, that was a response to one of the comments on the Bill Shillito video
this is gold
thanky for video, its very good
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.
It really helps me
Lots of love from bengal
Conics by Keith Kendig
Is where people should go after watching this awesome video
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.
You should submit this to SoME2! Awesome exposition on projective geometry
Thank you! It has already been submitted.
can you please do a video about affine geometry ?
Hello, I'm sorry to say that I won't be active for a while, and if I make another video it will most likely be unrelated to geometry (I am busy with college and two research projects at the moment).
I will say, however, that affine transformations in a n-dimensional space are equivalent to linear transformations in (n+1)-dimensional space that preserve some n-dimensional hyperplane that does not pass through the origin. For example, the affine transformation (x,y) --> (ax+by+e,cx+dy+f) that transforms the points (x,y) in 2D space can be turned into the linear transformation (x,y,z) --> (ax+by+ez,cx+dy+fz,z) that transforms the points (x,y,1) in the z=1 plane within 3D space. Theoretically, affine geometry could be formulated as a restricted version of linear algebra. Hope that helps!
@@coolcomputery
Wow, very informative; thank you so much for this little help, man !!!!!
Though I’d only just recommend that you use the symbol “|-->” for symbolizing (the action of) your function, when you’re describing it in terms of the general (set theoretical) form of your 3-tuplets; at least in the future
Also, another question : Whenever you used, such as (if that wasn’t the only time this _actually_ occurred as I described it) @…, as many rows in your vectors as there were dimensions in your vector space using them, well, since of course you can actually have many wildly more diversely “n-valued” n-tuplets that can be part of a vector space of a constant (i.e. definitely NOT ALWAYS “n-valued”, as opposed to what may imply its elements of theirs) dimensionality, I was wondering if it wasn’t actually that you were implicitly using some sort of “canon” or “ultimately simplified numeric & symbolic representation of the mathematical ideal of the vector space in question”, where you “reduce”, or “ _’extract’_ all of the _’redundant’_ _’numbers’_ & _’distances’_ between them”, or smth;
It especially hit me in how it looked like what basically boils down to, as far as I’ve actually read into the subject/topic, how a so-called “lossless image file data compression algorithm” would basically reduce the amount of information initially needed/procured by the initial file to properly record a certain pixel’s state (which’d also be structured computationally & logistically as an n-tuplet containing all the quantified properties as values arranged inside each of the “n” “slots” of that “tuplet” : The algorithm would, like, essentially reduce the cardinality of what’s essentilly just a family - i.e. “vector” or “tuplet”; or, again, it’d just lower the value of the “n” -, as well as just “bring all the numbers closer together”, with all the numbers - that are both/either possible &/or actually present in the currently processed/compressed file - occupying as little space inside the memory as possible)
👍
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?
the sound is weird great video
That IS how projective geometry was discovered. Just not how it was _rediscovered_ in Europe.
I'm not sure the word intuitive applies, by the end you defended into a bunch of symbolic manipulation that was not helpful to understanding.
True, the second half of the video was heavier on linear algebra and a way of manipulating matrices that might not be very well known, but I hope the first few minutes where I define projective points and lines were understandable and showed where projective geometry could have arisen from.