Can a bicubic Bezier surface be represented as a Bernstein polynomial? A Bernstein polynomial seems to simplify the representation of a 1-D Bezier curve. Does it work for a surface? And does this simplify the ray-surface intersection problem?
 A: Yes, if you have a parametric surface with equation $(u,v) \mapsto \mathbf{S}(u,v)$, and $\mathbf{S}(u,v)$ is a polynomial in $u$ and $v$, then it can be represented in Bezier-Bernstein form. The equation will take the form
$$
\mathbf{S}(u,v) = \sum_{i=0}^m  \sum_{j=0}^n \phi_i^m(u)\phi_j^n(v)\mathbf{P}_{ij}
$$
where the $\mathbf{P}_{ij}$ are control points and the $\phi$ functions are Bernstein polynomials.
See these notes or here for further info.
When you represent a surface in Bezier form, you haven't really changed anything; from a mathematical point of view, you've just done a change of basis. Whatever basis you use, you're still going to need numerical methods to solve the ray-surface intersection problem. However, working with the Bezier representation does have some benefits. First, numerical stability is very good. Secondly, the Bezier form allows you to do subdivison easily, and the convex hull property allows you to quickly eliminate portions of he surface, so you can use subdivision methods to do the intersection calculation. See here for some discussion.
