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1.4.4 B-spline surface

The surface analogue of the B-spline curve is the B-spline surface (patch). This is a tensor product surface defined by a topologically rectangular set of control points $ {\bf p}_{ij}$ , $ 0\leq i \leq m$ , $ 0\leq j \leq
n$ and two knot vectors $ {\bf U}=(u_0, u_1,\ldots,u_{m+k})$ and $ {\bf
V}=(v_0, v_1,\ldots,v_{n+l})$ associated with each parameter $ u$ , $ v$ . The corresponding integral B-spline surface is given by
$\displaystyle {\bf r}(u,v) = \sum_{i=0}^m\sum_{j=0}^n{\bf p}_{ij}N_{i,k}(u)N_{j,l}(v)\;.$     (1.86)

Parametric lines on a B-spline surface are obtained by letting $ u=const$ , or $ v=const$ . A parametric line with $ u=u_0$ is a B-spline curve in $ v$ with $ {\bf V}$ as its knot vector and vertices $ {\bf q}_j$ , $ 0\leq j \leq
n$ given by $ {\bf q}_j = \sum_{i=0}^{m}{\bf
p}_{ij}N_{i,m}(u_0)$ .

Some of the properties of the B-spline curves can be easily extended to surfaces such as:

However, no variation diminishing property is known for B-spline surface patches.


next up previous contents index
Next: 1.5 Generalization of B-spline Up: 1.4 B-spline curves and Previous: 1.4.3 Algorithms for B-spline   Contents   Index
December 2009