Efficient evaluation of multivariate polynomials. (English) Zbl 0606.65007

The authors give an algorithm to evaluate a polynomial of total degree d defined on a triangle T in the plane, \[ p(r,s,t)=\sum^{d}_{i=0}\sum^{i}_{j=0}c_{d-i,i-j,j}\cdot r^{d- i}s^{i-j}t^ j, \] where \(c_{d-i,i-j,j}=(d!/(d-i)!(i-j)!j!)b_{d- i,i-j,j}\), \(0\leq j\leq i\), \(0\leq i\leq d\), and (r,s,t) are the barycentric coordinates of each point in T, and \(b_{ijk}\) are the coefficients in the algorithm of de Casteljau. This algorithm is significantly faster than de Casteljau.
Reviewer: A.López-Carmona


65D10 Numerical smoothing, curve fitting
41A10 Approximation by polynomials
65D20 Computation of special functions and constants, construction of tables
41A63 Multidimensional problems
Full Text: DOI


[1] Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer Aided Geometric Design, 1, 1-60 (1984) · Zbl 0604.65005
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[3] Schumaker, L. L., Numerical aspects of spaces of piecewise polynomials on triangulations, (Mason, J., Proceedings of Shrivenham Conference on Approximation Theory (1986)) · Zbl 0628.65008
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