A unifying structure for polar forms and for Bernstein Bézier curves

Abstract

We construct polar forms for diverse types of spaces, including trigonometric polynomials, hyperbolic polynomials and special Müntz spaces, by altering the diagonal property of the polar form for homogeneous polynomials. We use this polar form to develop recursive evaluation algorithms and subdivision procedures for the corresponding Bernstein Bézier curves. We also derive identities and properties of these Bernstein bases and Bernstein Bézier curves, including affine invariance, curvilinear precision, end point interpolation, a degree elevation formula, a differentiation formula, a Marsden identity, a convex hull property, total positivity, and the variation diminishing property.

A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

“…Since the most of conic sections cannot be accurately represented by polynomials in explicit form, the parameter polynomials are used to approximate the conic sections. Bézier curves and surfaces [1][2][3][4] are the modeling tools widely used in CAD/CAM systems. Most of the previous work on conic sections approximation is based on quartic Bézier curves.…”

A functional generalization of the interpolation problem

“…. ; n are linearly independent on ½a; b (see [1]). We consider interpolation problem in the space p n ðf 1 ; f 2 Þ ¼ span / n 0 ; / n 1 ; .…”

“…The barycentric coordinates of a point PðxÞ ¼ ðf 1 ðxÞ; f 2 ðxÞÞ; a 6 x 6 b on this curve relative to the endpoints of the arc of PðtÞ from . It is shown in [1] that every element from the space p n ðf 1 ; f 2 Þ can be represented in terms of the barycentric coordinates.…”

“…(7), gðxÞ may not be an element of the space p n ðf 1 ; f 2 Þ. In [1] it is shown that if 1 2 spanff 1 ðxÞ; f 2 ðxÞg, then the barycentric coordinates sum to one, that is aðx; a; bÞ þ bðx; a; bÞ ¼ dðx; bÞ dða; bÞ þ dða; xÞ dða; bÞ ¼ 1:…”