What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces?

Question

I'm interested in problems involving geometric constraints and curve subdivision. I noticed that most of these problems describe the curves/surfaces using the Bezier form. I wanted to know if there are alternative ways of expressing an n-dimensional surface using control points that isn't a type of spline or Bezier curve.

Answer 1

Bezier curves are just polynomials. From a mathematical point of view, the $m+1$ Bernstein polynomials of degree $m$ are just a basis for the vector space $\mathbb{P}_m$ of all polynomials of degree $m$. So, of course, you can use other bases of $\mathbb{P}_m$, instead. This won't give you new types of curves and surfaces, just a different way of looking at them. Two other common choices are the "power" basis $\{1, u, u^2, \ldots, u^m\}$ and various Lagrange bases. Occasionally Legendre or Chebyshev polynomials. The Bernstein basis has some very attractive qualities. For example, it's very stable, numerically, it forms a partition of unity, so you get the convex hull property, and the coefficients (i.e. "control points") make some sense geometrically.

Also, regardless of what basis you use, polynomials are very attractive: easy to differentiate and integrate, easy to bound, useful for approximation, and generally well understood. Using anything else is going to be like swimming upstream, by comparison. Even using rational functions (as in rational Bezier curves) makes things quite a bit more difficult.