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Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming

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Abstract

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual functionals for the G-Bernstein basis functions and we use this dual functional property to prove that G-Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G-Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G-Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-Bézier curve, and we derive the geometric rate of convergence of this algorithm.

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Authors and Affiliations

  1. Department of Computer Science, Rice University, Houston, TX, 77251, USA

    Ron Goldman

  2. Department of Mathematics and Statistics, University of Houston-Downtown, Houston, TX, 77002, USA

    Plamen Simeonov

Authors
  1. Ron Goldman
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  2. Plamen Simeonov
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Correspondence to Plamen Simeonov.

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Goldman, R., Simeonov, P. Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming. Numer Algor 72, 605–634 (2016). https://doi.org/10.1007/s11075-015-0059-6

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