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https://en.wikipedia.org/wiki/Projective_geometry

https://en.wikipedia.org/wiki/Projective_geometry#/media/File:Theoreme_fondamental_geometrie_projective.PNG

I can certainly see that projection is used in the image, but it goes completely unexplained (at least on this page.) Meanwhile the "theory of projective geometry" is also unexplained and not mentioned again on the page. If you look it up elsewhere it doesn't appear (to me) to relate to the image, and doing an image search produces all kinds of other projections.

Looking at it myself, the last projection onto the green lines on the right makes no sense to me. Besides intersecting with teal, green looks like it's oriented arbitrarily, so I just don't see how the white lines are related. There is some text below the image but I must be dense. It's not intuitive enough that I would use it as an example to introduce newcomers to the concept and certainly not without explaining a thing. But I digress.

The fundamental theorem of projective geometry says that an abstract automorphism of the set of lines in Kn which preserves “incidence relations” must have a simple algebraic form.

Yep, nothing says "simple algebraic form" like twenty multicolored lines all intersecting with one another.

John P
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  • The bottom line is that, as ordinary Euclidean geometry is an axiomatization of, more-or-less, a real finite-dimensional vector space with a dot product, the projective geometry is the same but without the dot product. In Euclidean geometry, the transformations (isometries) preserve the dot product, and are orthogonal transformations from $O(n,\mathbb R)$. In projective geometry, the transformations are just any invertible linear operators from $GL(n,\mathbb R)$, which is a larger set. Colinearity is still preserved, angles are not (in fact, they are not even defined)... –  Sep 05 '22 at 22:54
  • ... but some things can still be defined and are invariant under those transformations, e.g. the four-ratio of four points on the same line. Also, conics can be defined and there are plenty of beautiful theorems that apply to them. (Except - you cannot tell a circle from an ellipse, from a parabola, from a hyperbola). –  Sep 05 '22 at 22:56
  • And that's what that image says to you? There's a slim chance I understood that... is it that the image depicts invertible linear operators from PG that aren't orthogonal, and that's why I'm confused? In any case, I dug a little more and found the theory it mentions in French, and there isn't an English version. I recognize cognates all over the place, but I'm beginning to think I wouldn't get it if I could read the rest... what, no angles?? – John P Sep 05 '22 at 23:18
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    The picture doesn't tell me anything more than it tells you. But I was lucky to have attended a projective geometry course many years ago, with some of it still lingering in my brain... –  Sep 05 '22 at 23:21
  • I believe the picture shows some $1$-dimensional projective transformations on a line and illustrates the fact that a projective transformation is purely defined on three points, i.e. if a projective transformation of a line keeps three points fixed, it is an identity. –  Sep 05 '22 at 23:31
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    It's amazing and shocking to find that there is a Wikipedia article on mathematics which is nearly incomprehensible, contains horrible, overcomplicated, unillustrative graphics, and is written in a dense, overabstract way such that nobody who is not already an expert can possibly understand it. – Suzu Hirose Sep 05 '22 at 23:57
  • Do you understand graphical projection? Like onto vanishing points? – CyclotomicField Sep 06 '22 at 00:04
  • @SuzuHirose Sarcasm of the year award – John P Sep 06 '22 at 02:07
  • Oh good lord. You're going to love the translated version of the PG fundamental theory. https://fr-m-wikipedia-org.translate.goog/wiki/Th%C3%A9or%C3%A8me_fondamental_de_la_g%C3%A9om%C3%A9trie_projective?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp "Let K be a field (commutative or not) and E and F vector spaces of finite dimension n greater than or equal to 2 over K. We call projective frame of the projective space P ( E ) deduced from E any ( n + 1)-tuple of points of P ( E ) such that none of these points belongs to the projective subspace of P ( E ) ) (...)" Who is this even for? – John P Sep 06 '22 at 02:11

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How does this image demonstrate the fundamental theory of projective geometry?

Short answer: it doesn't.

The image illustrates two sequences of central projections from one line $d$ to another line $f$. The diagram keeps track of where three specific points get projected. In particular, the two chains, $d\to e\to f$ and $d\to g\to h\to f$ result in the same the points $A_3,B_3,C_3$. Presumably the relevance of this is to then argue / demonstrate / discuss that if the two chains agree in how they transform three distinct points, then they will agree in how they transform any other point, too. But this is not apparent from the picture alone.

The image appears to have some legend in French, so presumably the force for it is the French Wikipedia. There it occurs in an article about the fundamental theorem of projective geometry. Note it's*theorem," not theory in the wording. The English Wikipedia has two articles on this, https://en.wikipedia.org/wiki/Fundamental_theorem_of_projective_geometry redirects to https://en.wikipedia.org/wiki/Homography#Fundamental_theorem_of_projective_geometry but there is also https://en.wikipedia.org/wiki/Collineation#Fundamental_theorem_of_projective_geometry with the two referencing one another. So if you want an explanation of that fundamental theorem, here they are. However, it's just a name, and at least in my opinion the fundamental theorem in projective geometry is not as central as fundamental theorems in other fields might be. You can do a lot of projective geometry without ever encountering that theorem.

Even the tie between the image and the theorem isn't particularly strong. It's not like the image is supposed to illustrate the theorem in its entirety. In the French article it's apparently illustrating one specific property which that article treats as an axiom. Namely that the only projective transformation of the line with three fixed points is the identity. Since none of the other axioms even speak about the concept of a projective transformation, that feels weird, too.

MvG
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