B-spline surface and its advantages

A B-spline surface is a tensor-product surface built from B-spline basis functions in two directions. It is defined by a control net and two knot vectors.

Equation

For an $(m+1)\times(n+1)$ net $\mathbf B_{ij}$, with orders $k$ in $u$ and $l$ in $w$:

$$\mathbf P(u,w)=\sum_{i=0}^{m}\sum_{j=0}^{n}\mathbf B_{ij}\,N_{i,k}(u)\,M_{j,l}(w)$$

$N_{i,k}$ and $M_{j,l}$ are B-spline basis functions defined by the knot vectors $U$ and $W$ (Cox-de Boor recursion). A bicubic B-spline surface uses $k=l=4$.

Comparison

Advantages

1. A large, complex surface (car roof, wing) is one entity with a low degree.
2. Local modification does not disturb the rest of the surface.
3. Smooth ($C^2$ for bicubic) without extra constraints.
4. Convex hull property and affine invariance hold.
5. The rational version (NURBS) also represents exact cylinders, cones, spheres and tori, and is the standard in CAD and IGES/STEP exchange.