Desargues Theorem and help with its significance

Question

I am still trying to get a hang of drawing the picture. The only idea I get from the theorem is that if two triangles are in perspective from a point, then we the theorem , we also get that the triangles are perspective from a line as well. How can I connect this with the idea of projective geometry? Or is it affine geometry? What else is this theorem saying and how and why is Pappus theorem related and important as well?

Answer 1

Desargues' theorem is not quite generally true in the affine plane, but it the projective plane it has no exceptions.

Say two triangles are in perspective from a point. The theorem says then they are in perspective from some line, BUT in some cases it turns out that either that is the line at infinity or one of the three points of interest on it is a point at infinity. Say the vertices of one triangle are $a,b,c$ and those of the other are $A,B,C$. Then the lines $Aa$, $Bb$, $Cc$ meet at the center of perspectivity. So the lines $AB$ and $ab$ meet at a point $X$, right? Not quite: they may be parallel, so $X$ is at infinity, but of course the affine plane has no points at infinity. Suppose that happens and the lines $AC$ and $ac$ meet at a finite point $Y$ and the lines $BC$ and $bc$ meet at a finite point $Z$. The theorem says $X,Y,Z$ should be colinear. What does it mean to say they're colinear when $X$ is a point at infinity? It means that $X$ is the one particular point at infinity that is on the line $YZ$. That's the same as saying the line $YZ$ is parallel to the lines $AB$ and $ab$.

It can also happen that all three, $X$, $Y$, and $Z$ are on the line at infinity. (But you can't have just two of the at infinity and the third a finite point, since then they would not be colinear.)

If two triangles are in perspective from a point, that point itself may be at infinity, meaning the lines $Aa$, $Bb$, $Cc$ are parallel to each other.

So if there are no points at infinity, as in the affine plane, then Desargues' theorem has exceptions.

In the projective plane there are no exceptions. In the projective plane you can pick ANY line and call it the line at infinity and delete it and whatever is left is an affine plane. So you don't need to mention anything about lines or points at infinity when stating Desargues' theorem in the projective plane.