18 points ibobev 3 hours ago 7 comments
nh23423fefe 1 hour ago | parent
The only time i've understood applications of projective geometry was seeing rotations and translations unified using exponentials of bivectors. I still dont get homogenous coords. Like i get how f(lambda*x)=lambda^n*f(x) and that means f is like a homomorphism but i've never solved any problem sets and understood why one would do this in the first place. I get what stereographic projection is, but can't apply it to problems.
whereas i constantly see applications of other math objects i've learned about. i never hear myself think, 'oooh yeah thats projective and so...'
pjbk 52 minutes ago | parent
Makes an affine space linear. It was a natural extension to provide direct transformations that encapsulate symmetry transformations, introduced by Mobius in his barycentric calculus.
QuesnayJr 22 minutes ago | parent
A general affine coordinate change in 3d is Ax + b, where A is a 3-by-3 matrix and b is a 3d vector. If you add a fourth coordinate that you set to 1, you can write it as a 4-by-4 matrix. That's homogeneous coordinates.
abetusk 1 hour ago | parent
The motivation is better than most, trying to treat infinity as just another case instead of an exceptional one, but, to me at least, it still feels like a "hack" to put in a third dimension and then insist on dividing by it to reproject whenever you want to do anything useful.
I'm a little used to the trick but it'd be nice to give some deeper motivation of why the trick works in the first place and why this particular representation.
ArcHound 34 minutes ago | parent
Lemme try. Have you ever seen a railroad vanishing in the horizon? You know the rails never meet and yet, they definitely meet in infinity before your very eyes. That's because you see a 2D projection of a 3D space. In 3D, you'll never find the intersection. It gives one of the best intuitions on how to work with points in infinity.
edflsafoiewq 17 minutes ago | parent
If you want to study how a polynomial P(X,Y) behaves at infinity, you can't set eg X,Y=infinity, but you can take P(X/W,Y/W) when W goes to zero. Now clear denominators by multiplying by an appropriate power of W and you get a homogeneous polynomial in X,Y,W. You can now set W=0 directly. Scaling a solution cX,cY,cW doesn't change the original solution since (cX)/(cW) = X/W, etc.
I guess the important part is clearing denominators turns problems with division into linear problems.
QuesnayJr 25 minutes ago | parent
The problem with math is that there's more you need to know than there's class time to teach it to. Projective space is a good example. It comes up in several different ways, but you're probably not going to see them all.