Thursday, March 05, 2009

Desargues' Theorem

My geometry class this semester seems to revolve around Desargues' Theorem (pronounced dezargs), which is a slightly complicated theorem that doesn't hold in the Euclidean plane, but holds in a lot of projective spaces. It took me a long time to be really comfortable with it, but it now seems quite simple to me, probably because of the dozens of times that I have drawn and explained it.

Desargues' says, simply, that if two triangles are perspective from a point, they are also perspective from a line.

"Perspective from a point" is relatively simple, though it took me a while to learn what motivated calling it that.

In this picture, the two triangles ABC and A'B'C' are perspective from V because if you were looking from V (hint: like the giant eye), the corresponding points of the triangles would be on the same lines. (Here, "triangle" actually refers to the points, not the line segments or the interior, though it's not very important in this case.)

"Perspective from a line," alas, doesn't allow quite such a simple explanation. What it means is that the intersections of the corresponding sides (e.g., where AB crosses A'B', if you extend the line segments) are collinear, or all on the same line. Let's expand the drawing above to include those points and their (possible) line of perspective. (Note: I've also moved the points around slightly to make the intersections show up.)

Here, I've colored the corresponding sides the same color, so AB and A'B' are blue, AC and A'C' are red, and BC and B'C' are green. Where each pair intersects, you get the purple points L, M, and N, and the big purple line goes through all of them. So, in this case, Desargues' Theorem does hold: ABC and A'B'C' are perspective from both V and the the purple line.

The only reason Desargues' doesn't always hold in a Euclidean plane (the kind of geometry we're used to) is that some of the lines might be parallel. For instance, if AC and A'C' were parallel, N (their intersection) wouldn't exist. Projective geometry basically does away with parallel lines, so it doesn't have that limitation.

1 comment:

Sally said...

Oooooh, pretty pictures! (Really, quite nice. The giant Egyptian eye is a lovely touch.)

Oh yeah, and there also appears to be geometry involved.