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On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus (III): Blossoming ☆

Research paper by Rudolf Winkel

Indexed on: 13 May '16Published on: 11 May '16Published in: Computer Aided Geometric Design



Abstract

The investigation of <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-Bernstein polynomials and <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-Bézier curves is continued in this paper. It is shown that convolution of the parameters <img height="13" border="0" style="vertical-align:bottom" width="93" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si2.gif">a¯=(a¯1,…,a¯n) is fundamental for (1) the definition of <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-Bernstein polynomials, (2) a simplified derivation of the <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-de Casteljau algorithm, (3) the recurrences that give the blossoming of <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-Bernstein polynomials and <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-Bézier curves, (4) the dual functional property and the <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-dual functional property for an <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-Bézier curve – it is necessary to make this distinction – and (5) the <img height="11" border="0" style="vertical-align:bottom" width="9" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167839616300486-si1.gif">a¯-degree elevation.