Elementary question which I sort of skipped and am now banging my head on: given the equation for a general 2- sphere $$(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$$
I need to construct a parametrization by the complex projective line.
I've just had a total conceptual block on this - I can do the algebra necessary to get it so that we have a point at the affine origin of co-ordinates in $\mathbb{A}^3$, but I just can't really get any intuition for what a complex homogeneous co-ordinate $(s:t)$ is really doing.
The problem I'm having is that I've got four variables to play with and only three that need to be parametrized. Or have I got the wrong end of the stick here?