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An algorithm for direct multiplication of b-splines.
| Content Provider | CiteSeerX |
|---|---|
| Author | Riesenfeld, Richard F. Chen, Xianming Cohen, Elaine |
| Abstract | B-spline multiplication, that is, finding the coefficients of the product B-spline of two given B-splines is useful as an end result, in addition to being an important prerequisite component to many other symbolic computation operations on B-splines. Algorithms for B-spline multiplication standardly use indirect approaches such as nodal interpolation or computing the product of each set of polynomial pieces using various bases. The original direct approach is complicated. B-spline blossoming provides another direct approach that can be straightforwardly translated from mathematical equation to implementation; however, the algorithm does not scale well with degree or dimension of the subject tensor product B-splines. To addresses the difficulties mentioned heretofore, we present the Sliding Windows Algorithm (SWA), a new blossoming based algorithm for the multiplication of two B-spline curves, two B-spline surfaces, or any two general multivariate B-splines. Note to Practitioners: Geometric kernels in commercial CAD systems typically use B-splines to represent smooth curves and surfaces. Geometric inquiry (such as curvature) on such curves and surfaces requires the fundamental mathematical operation of multiplying two B-splines. There are a few existing algorithms in the CAD community to perform Bspline multiplication. All of them are indirect methods, in the sense of either by some sampling and interpolation strategy, or leaving the domain of B-spline representation. The only direct multiplication, reported in early 1990s, actually only solved the problem from a purely mathematical perspective. It is so inefficient as to be not feasible for any practical usage. The presented paper re-exams this initial idea of direct Bspline multiplication, and finds some simple characteristics of the apparently combinatorial problem, and designs a set of efficient algorithms, known as Sliding Window Algorithm (SWA). |
| File Format | |
| Access Restriction | Open |
| Subject Keyword | Practical Usage Fundamental Mathematical Operation Bspline Multiplication Initial Idea Indirect Approach Commercial Cad System New Blossoming Presented Paper Re-exams Direct Approach Window Algorithm Important Prerequisite Component Polynomial Piece B-spline Representation Direct Multiplication Product B-spline Indirect Method Interpolation Strategy Cad Community Abstract B-spline Multiplication B-spline Curve Simple Characteristic End Result Mathematical Perspective Geometric Inquiry Subject Tensor Product B-splines B-spline Multiplication Efficient Algorithm General Multivariate B-splines Sliding Window Algorithm Geometric Kernel Various Base Smooth Curve Mathematical Equation Many Symbolic Computation Operation Nodal Interpolation Direct Bspline Multiplication Index Term Nurbs Multiplication Original Direct Approach Combinatorial Problem B-spline Blossoming B-spline Surface |
| Content Type | Text |
