Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters

Abstract

Abstract A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression is constructed that produces the generalized blended trigonometric Bernstein (or GBT-Bernstein, for short) polynomial functions of degree m. Using these basis functions, generalized blended trigonometric Bézier (or GBT-Bézier, for short) curves with two shape parameters are also constructed, and their geometric features and applications to curve modeling are discussed. The newly created curves share all geometric properties of Bézier curves except the shape modification property, which is superior to the classical Bézier. The C 3 $C^{3}$ and G 2 $G^{2}$ continuity conditions of two pieces of GBT-Bézier curves are also part of this study. Moreover, in contrast with Bézier curves, our generalization gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling.

Keywords

• GBT-Bernstein-like polynomial functions
• GBT-Bézier curve
• Properties of GBT-Bézier curves
• Continuities of GBT-Bézier curves
• Shape parameters