Quantum Trigonometric Bézier Curves

TL;DR

The paper develops quantum trigonometric Bézier curves by introducing a one-parameter family of quantum trig Bernstein bases with a shape parameter $q$, situating them in the trigonometric polynomial space $T_n$ and analyzing total positivity to infer shape properties. It provides two de Casteljau-type recursive evaluation algorithms for these curves and shows how the rational extension, using positive weights, inherits strong shape-preserving properties. Specifically, the rational bases are normalized totally positive on subintervals $[rac{k p i}{2},rac{(k+1) p i}{2}]$ and yield end point interpolation, convex hull containment, variation diminishing, and affine invariance under $q>0$ and $w_k>0$. The work also discusses subdivision considerations and outlines future work to obtain curve-segment control points via extended de Casteljau techniques.

Abstract

In order to construct quantum trigonometric Bézier curves with shape parameter, one parameter family of trigonometric Bernstein basis functions are introduced. We study the total positivity of the basis functions to analyze the shape preserving properties of the quantum trigonometric Bézier curves. We also showed that quantum trigonometric Bézier curves can be evaluated by two different recursive evaluation algorithms. Finally, we have defined rational counterpart of quantum trigonometric Bézier curves and show that the rational quantum trigonometric Bézier curves posses nice shape preserving properties.

Figures (4)

• Figure 1: Cubic quantum trigonometric basis functions on $[a,b]=[\pi/8,\pi/4]$ for $q=1.1$ (blue), $q=1.2$ (green) and $q=1.3$ (red).
• Figure 2: Cubic quantum trigonometric basis functions on $[a,b]=[0,\pi/2]$ for $q=1.1$ (blue), $q=1.2$ (green) and $q=1.3$ (red).
• Figure 3: Third degree rational quantum trigonometric Bernstein bases, with $q=1.1$ (blue), $q=1.2$ (green) and $q=1.3$ (red), defined over the interval $\left[0,\frac{\pi}{2}\right]$ where all weights are equal to 1.
• Figure 4: Third degree rational quantum trigonometric Bézier curves associated with the control polygon. Here $\mathbf{b}_{0}=(0,0),$$\mathbf{b}_{1}=(1,2),$$\mathbf{b}_{2}=(2,2),$$\mathbf{b}_{3}=(3,0),$ all $w_{k}=1$ and $q=1$ (blue), $q=2$ (green) and $q=3$ (red). The rational quantum trigonometric Bézier curves are evaluated on the interval $\left[0,\frac{\pi}{2}\right].$

Theorems & Definitions (11)

• Definition 2.1
• Theorem 2.2
• proof
• Definition 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• Definition 4.1
• Definition 4.2
• Theorem 4.3