On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus |
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Authors: | Rudolf Winkel |
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Affiliation: | University of Applied Sciences Bingen, Berlinstr. 109, D-55411 Bingen on the Rhine, Germany |
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Abstract: | In Winkel (2001) a generalization of Bernstein polynomials and Bézier curves based on umbral calculus has been introduced. In the present paper we describe new geometric and algorithmic properties of this generalization including: (1) families of polynomials introduced by Stancu (1968) and Goldman (1985), i.e., families that include both Bernstein and Lagrange polynomial, are generalized in a new way, (2) a generalized de Casteljau algorithm is discussed, (3) an efficient evaluation of generalized Bézier curves through a linear transformation of the control polygon is described, (4) a simple criterion for endpoint tangentiality is established. |
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Keywords: | Umbral calculus Generalized Bernstein polynomial Generalized de Casteljau algorithm Interpolation Efficient evaluation |
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