projective space and homogenous coordinates.

Guess I've got the projective space right now. Love the explanation at wiki.
http://en.wikipedia.org/wiki/Projective_space
But its urgent to really understand that and the homogeneous coordinates.
Projective space: really a projection! 3d space projected into a tiny 2d picture. A projection from 3d to 2d.

Quote:'the set of equivalence classes of R3\(0, 0, 0), i.e. 3-space without the origin, where two points P = (x, y, z) and Pˈ = (xˈ, yˈ, zˈ) are equivalent if there is a nonzero real number λ such that P = λ·Pˈ, i.e. x = λxˈ, y = λyˈ, z = λzˈ. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point (x, y, z) in R3, is

[x : y : z].

The last formula goes under the name of homogeneous coordinates'

You can homogenize like this: you add z until all parts of the equation have the same degree:

y^2=x^3 + Ax + B becomes y^2z=x^3 + Axz^2+Bz^3

Calculating in projective space gives us a way to work with the point at infinity without just panicking. The point at infinity just becomes a special plane i a 3 dimensional space, where parallel lines meet.
lucky bastards.