Novel polynomial Bernstein bases and B,zier curves based on a general notion of polynomial blossoming

作者：Goldman Ron; Simeonov Plamen^*

来源：Numerical Algorithms, 2016, 72(3): 605-634.

DOI：10.1007/s11075-015-0059-6

• G-blossom
• G-Bernstein basis
• G-Bezier curve
• Recursive evaluation algorithm
• Dual functional property
• Marsden identity
• Polya curve
• Variation diminishing property
• Subdivision

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-B,zier 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-B,zier curves share several other properties with classical B,zier 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-B,zier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-B,zier curves are of Plya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-B,zier curve, and we derive the geometric rate of convergence of this algorithm.

• 出版日期2016-7