I'm familiar with projective coordinates for elliptic curves, but the funny thing about curves is that, for high-speed software, the math is tricky enough that you don't "need" to grok it: there's an "explicit formulas database" that you can just copy from:

http://www.hyperelliptic.org/EFD/g1p/index.html

Curves are what made algebra 'click' for me, getting me from my high school understanding of "algebra is math about unknown variables" to "algebra is about sets of related objects with operators that have identities and inverses".