Degree elevation of weighted Bezier curve

Question

I'm having difficulty understanding the derivation of the formula for degree elevation of a weighted Bezier curve given here. The only information that's given is to project a Bezier curve info affine space in order to obtain the desired expression. I'd appreciate help understanding this "derivation".

Answer 1

I'll assume that you understand how to do degree elevation for a polynomial curve. If so, go though this same calculation, but use 4D control points $\mathbf{Q}_i = (w_ix_i, w_iy_i, w_iz_i, w_i)$ instead of 3D points $\mathbf{P}_i = (x_i, y_i, z_i)$. The algebra is exactly the same, regardless of whether you use 3D points or 4D ones. Then, once you have the 4D formula, project it back to 3D using the standard map $(x,y,z,w) \mapsto (x/w,y/w,z/w)$. This last step is what Farin means when he says "project into affine space".