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

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.

Geometric modeling of some engineering GBT-Bézier surfaces with shape parameters and their applications

“…This research work is mainly focused on the construction of engineering surfaces in Bézier approach. For this purpose, utilizing GBT-Bézier curves proposed in [24], six different types of tensor product GBT-Bézier engineering surfaces including GBT-Bézier cylinders, GBT-Bézier bilinear surfaces, GBT-Bézier ruled surfaces, GBT-Bézier swung surfaces, GBT-Bézier swept, and GBT-Bézier rotation surfaces are constructed. All geometric features of proposed GBT-Bézier surfaces are similar to the traditional Bézier, but the shape-adjustment feature is an additional feature that is not present in the traditional Bézier surface approach.…”

“…are GBTB basis functions of order m [24]. In particular, when k = -1 or k > n, the functions f k,m (z) = 0.…”

“…The GBT-Bézier curves take over most advantageous features of the traditional Bézier curves, including symmetry, convex hull property, variation diminishing property, geometric invariance, shape adjustable property, and the terminal properties F(0; μ, ν), F(1; μ, ν), F (0; μ, ν), F (1; μ, ν), F (0; μ, ν), F (1; μ, ν), F (0; μ, ν), and F (1; μ, ν). The terminal properties of GBT-Bézier curves indicate that the GBT-Bézier curve segments interpolate to the both end points of their convex hull, and the values of their derivatives at both end points on GBT-Bézier curves will be easily modified by adjusting the shape parameters μ, ν in their respective value range, which brings a significant ease to smooth joining [24].…”

Simultaneous Developability of Partner Ruled Surfaces according to Darboux Frame in E 3

“…Our objective is to study the simultaneous developability of such couples of surfaces. Through our study, we are opening up some avenues for scientists to apply our approach in some areas such as architectural design, medical science, surface modeling, engineering, and computer-aided geometric design [12][13][14][15].…”

Typical curve with G1 constraints for curve completion

“…In recent years, geometric continuity has also been discussed based on trigonometric Bézier curves with shape parameters [30][31][32][33]. Generalized trigonometric Bézier curves are flexible and can achieve G2 continuity.…”