Introduction to Projective Geometry (Part 1)

[@johnrobin4569]

A great explanation on affine geometry, especially regarding how to generalize the concept of line/pt using the 4 rules. I am wondering about one thing in the 4-point example. For a line joining two points diagonally, how would we define the non-intersecting unique line passing through the 3rd point? Assuming there are only 4 points, there are only the 6 lines defined and hence only the other diagonal is possible. Am I right to assume that the diagonal lines do not "cross" (in the sense that there is no actual point) and that instead the idea of "line" here is defined as a connection between points?

[@k-theory8604]

Yes, that's exactly right! Though they "intersect" when literally drawing the lines connecting them, they don't represent intersecting lines in the geometry, because the only points are the 4 corners. In this sense, drawing literal lines connecting them is somewhat flawed, since we're used to R^3 where every dot of ink on a given line represents a point. In some sense a more accurate picture here would be to list the lines as sets. For examples, if the vertices were labelled clock-wise as 1,2,3, and 4, then our plane is really just the collection of "lines" {1,2}, {2,3}, {3,4}, {4,1}, {1,3}, {2,4}

[@francescominnocci]

@@k-theory8604 In this context the fact that the two non-intersecting lines are parallel (for rule #3) is equivalent to the fact that they do not intersect(?)

[@k-theory8604]

@@francescominnocci yes.

[@melissapereira7654]

can you please do a video about affine geometry?

[@k-theory8604]

I would like to eventually. However I'm already far behind on a couple of series I had intended to finish by now, so it may take some time.

[@kareemakram3676]

is it the same as descriptive geometry

[@k-theory8604]

It's not the same as descriptive geometry, but it certainly is related to it! Ideas from projective geometry can be used in descriptive geometry, and historically, projective geometry arose from an early version of descriptive geometry.

[@鸞麤]

in the example you gave of a finite affine geometry, the line connecting b and c does not satisfy axiom 3

[@k-theory8604]

Remember that the only "points" I'm considering are the 4 red dots, and "lines" are only to be though of as collections of the red dots. The points on the line in my drawing do not count as "points" of this particular geometry, so even though the pair of lines a-c and b-d appear to intersect in the middle of the drawing, they do not. If it makes it easier for you to think about, you might want to consider the image as being in 3D space, so the lines representing this geometry don't literally intersect in 3D space. So let's think about the line b-c. To see that it satisfies axiom 3, we need to consider the points a and d. First, note that a lies on exatly 3 lines a-b, a-c, and a-d. The only one of these that doesn't intersect b-c is the line a-d. Hence the axiom is satisfied. Likewise, d lies on exactly the lines d-c, d-b, and d-a. The only one of these that does not intersect b-c is the line d-a. Thus we see that in either case, axiom 3 is satisfied.

[@鸞麤]

​@@k-theory8604 ah yeah, of course, that makes sense. i see where i went wrong. thanks for clearing that up.

[@AnonTen]

colliner 💀