Abstract
In this paper, properties and algorithms of q-Bézier curves and surfaces are analyzed. It is proven that the only q-Bézier and rational q-Bézier curves satisfying the boundary tangent property are the Bézier and rational Bézier curves, respectively. Evaluation algorithms formed by steps in barycentric form for rational q-Bézier curves and surfaces are provided.
Highlights
The q-Bernstein basis of polynomials for some 0 < q ≤ 1 plays an important role in several areas, such as Approximation Theory and Computer Aided Design
We prove that the Bernstein basis is the unique basis among all q-Bernstein bases satisfying the geometric boundary tangent property
In this subsection we prove that the boundary tangent property holds for q-Bézier curves if and only if q = 1, that is, for Bézier curves
Summary
The q-Bernstein basis of polynomials for some 0 < q ≤ 1 (see [1]) plays an important role in several areas, such as Approximation Theory and Computer Aided Design. In Computer-Aided Geometric Design (CAGD), it is desirable that a basis U satisfies the endpoint interpolation property and the boundary tangent property. It is well known in CAGD that a curve representation presents nice properties when the corresponding normalized basis is totally positive, that is, when all its collocation matrices have nonnegative minors (see [4,5]). It is well known that Bézier curves satisfy the endpoint interpolation property and the boundary tangent property. In [8] it was shown that all rational q-Bernstein bases satisfy the endpoint interpolation property. We prove that the Bernstein basis is the unique basis among all q-Bernstein bases satisfying the geometric boundary tangent property.
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