
BSplines and Tensor Product BSplinesBsplines are fundamental to almost any application in geometric modeling. In particular, nonuniform rational Bspline representations (NURBS) have become a standard in CAD and CAM. Via the WEBmethod, Bsplines also provide very efficient finite element approximations. Bsplines are piecewise polynomial functions, which can be defined by a simple recursion. The following Figure shows Bsplines of order 2 to 4 (piecewise polynomial functions of degree 1 to 3) on a uniform partition. A Bspline of order n is positive and has compact support (marked red in the above Figure), consisting of n grid intervals. At the break points, the polynomial segments join with maximal smoothness, i.e., all but the (n1)st derivative match. Multivariate basis functions are constructed by multiplying Bsplines corresponding to the different coordinates.
These socalled tensor product Bsplines of order n also have compact support and are (n2)times continuously differentiable. Bsplines have the local approximation power of polynomials. Hence, high accuracy can be achieved already with relatively few basis functions. Author: Joachim Wipper ; Last modification: 2006/08/31 09:38:46 UTC.
