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- W2349169912 abstract "NURBS offers a common mathematical form for representing and designing both analytic and free form curves and surfaces. So it is specially important to seek the proper forms of representation for nonuniform B spline itself, which is the basis for NURBS curves and surfaces. The key to matrix representation for NURBS curves and surfaces is the piecewise power function base representation for B spline. In the view of algebra, the coefficient matrix of a NURBS curve or surface is just the transformation matrix of a B spline base to a power function base at each subinterval in a polynomial spline space. Therefore seeking explicit matrix representation for NURBS curves and surfaces is actually seeking the explicit piecewise polynomial representation for B spline base functions. In the past researches, B spline base functions were generally represented in the form of recursion or divided difference, which makes it difficult to obtain the explicit analytic representation for coefficient matrices of NURBS curves and surfaces. In this paper, the concept of divided difference expanded coefficient is presented and its explicit analytic representation formula is derived. Based on this formula and the divided difference definition of B spline, the explicit piecewise polynomial representation for B spline base functions is obtained and then an explicit matrix representation formula for NURBS curves and surfaces of arbitrary order is presented. Besides, efficient numerical algorithms for the calculations of divided difference expanded coefficient and the coefficient matrix of NURBS curves and surfaces are also given respectively. They can be easily realized. The results can be applied to all NURBS curves and surfaces including rational and nonrational Bezier, uniform and nonuniform B spline forms. The analytic representation formula of divided difference expanded coefficient provides a uniform constructive tool and application approach for the research of NURBS curves and surfaces such as their representation, transformation, knot insertion, degree elevation, etc." @default.
- W2349169912 created "2016-06-24" @default.
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- W2349169912 date "2001-01-01" @default.
- W2349169912 modified "2023-09-23" @default.
- W2349169912 title "Explicit Matrix Representation for NURBS Curves and Surfaces and Its Algorithm" @default.
- W2349169912 hasPublicationYear "2001" @default.
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