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...'

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.