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...'

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.

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.