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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 (should also be assigned at least one other classification number from Section 41-XX)
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
[2] de Casteljau, F., Courbes et surfaces à pôles, (1963), André Citröen Automobiles Paris
[3] Schumaker, L.L., Numerical aspects of spaces of piecewise polynomials on triangulations, () · Zbl 0628.65008
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