An ill-conditioned problem in computer aided geometric design
Neural, Parallel & Scientific Computations - computer aided geometric design
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Ray tracing parametric patches
SIGGRAPH '82 Proceedings of the 9th annual conference on Computer graphics and interactive techniques
Multivariate resultants in Bernstein basis
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
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The Berstein basis is used extensively in geometric modelling because of its elegant geometric properties and simple algorithms that are available for processing it. Although resultants are used for several important operations in geometric modelling and computer graphics, it is necessary to perform a polynomial basis transformation because the established theory of resultants assumes that the polynomials are expressed in the power (monomial) basis. In this paper, the numerical behavior of a resultant matrix for a scaled Bernstein polynomial (a polynomial of degree n whose basis functions are (1-x)n-ixi, i=0,,n is investigated. In particular, a companion matrix M for a scaled Bernstein polynomial r(x) is developed and this is used to form a resultant matrixs(M), where s(x) is a scaled Bernstein polynomial. Computational evidence is presented that suggests that this method of computing the resultant of two Bernstein basis polynomials is superior to the established method of using a simple parameter substitution to perform a change from the Bernstein basis to the power basis.