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.