Visualizing more projective curves
Here's another example of how an affine curve can have different properties than its corresponding projective curve. Here's what the affine curve y = x3 and the projective yz2 = x3 look like. One thing that stands out is how the projective curve has two singularities while the affine version is always nice and smooth.
The projective yz2 = x3 meets z = 0 at (0,1,0), or where y = 1. If we let y = 1 in yz2 = x3 then we're left with z2 = x3, so that we should expect yz2 = x3 to behave roughly like z2 = x3 out where z = 0. This causes the very sort of singularity that we see in these pictures.
Now I just have to figure out how to hack POV-Ray to show the projective form of an elliptic curve.