Curve and surface construction based on the generalized toric-Bernstein basis functions

Abstract

<jats:title>Abstract</jats:title> <jats:p>The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.</jats:p>

Total positivity of a kind of generalized Toric-Bernstein basis

“…Inspired by their work, Li et. al [26,27] defined a kind of GT-Bernstein basis using a set of real points and then applied it in the curve and surface design. 26,27]).…”

“…al [26,27] defined a kind of GT-Bernstein basis using a set of real points and then applied it in the curve and surface design. 26,27]). Let S = {a 0 , · · · , a n } ⊂ R be a finite set, where a 0 a 1 · · · a n−1 a n , and a 0 < a n .…”

“…In [27], they also construct the so called GT-Bézier surface by two dimensional GT-Bernstein basis. We conjecture that the rational toric-Bernstein basis defined on two (or higher)-dimensional real points is also NTP basis, and then GT-Bézier surface also has the PIA property.…”

“…Remark 2. Although a GT-Bernstein basis {β a i (t)} depends on the selection of l 0 , l 1 , the generalized toric-Bézier curve (GT-Bézier curve) defined by it depends only on the set S and independent of the selection of l 0 , l 1 [26,27]. In the rest of the paper, we assume that l 0 = l 1 = l > 0.…”

“…al [25] proposed a class of irrational toric varieties, inspired by which Li et. al [26,27] defined a kind of generalized toric-Bernstein basis (GT-Bernstein basis) using a set of real points, and then applied it in the design of curve/surface which is called generalized toric-Bézier curve/surface (GT-Bézier curve/surface). The main purpose of this paper is to prove that rational GT-Bernstein basis is NTP basis.…”