Abstract
Adopting a recurrence technique, generalized trigonometric basis (or GT-basis, for short) functions along with two shape parameters are formulated in this paper. These basis functions carry a lot of geometric features of classical Bernstein basis functions and maintain the shape of the curve and surface as well. The generalized trigonometric Bézier (or GT-Bézier, for short) curves and surfaces are defined on these basis functions and also analyze their geometric properties which are analogous to classical Bézier curves and surfaces. This analysis shows that the existence of shape parameters brings a convenience to adjust the shape of the curve and surface by simply modifying their values. These GT-Bézier curves meet the conditions required for parametric continuity (C0, C1, C2, and C3) as well as for geometric continuity (G0, G1, and G2). Furthermore, some curve and surface design applications have been discussed. The demonstrating examples clarify that the new curves and surfaces provide a flexible approach and mathematical sketch of Bézier curves and surfaces which make them a treasured way for the project of curve and surface modeling.
Highlights
Bezier curves and surfaces have been extensively used in computer graphics (CG) and computer-aided geometric design (CAGD) because of their valuable properties
Since traditional Bezier curves can be obtained by control points and Bernstein basis functions, after creating Bezier curves and Mathematical Problems in Engineering surfaces, we can construct different shapes by using parametric and geometric continuities which fulfill our design requirements
An extension of quartic Bezier curve with three shape parameters is presented by Zhu et al [13], which is a continuation to a forth degree Bezier curve with a single parameter which improved the shape control of the curve
Summary
We construct the GT-basis functions by using a recursive relation . are known as GT-basis functions of degree 2. e function wi,m(z)(i 0, 1, . . . , m) described recursively for any integer m(m ≥ 3) as wi,m(z) 1 −. We construct the GT-basis functions by using a recursive relation . Are known as GT-basis functions of degree 2. M) described recursively for any integer m(m ≥ 3) as wi,m(z) 1 −. Is GT-basis function of the mth order. In situation, when i − 1 or i > m, the function wi,m(z) 0. E GT-basis functions enjoy many properties as follows:. (2) Nonnegativity: for α, β ∈ [− 1, 1], wi,m(z) ≥ 0 E graphs of 5th and 10th degree GT-basis functions with shape parameters α, β − 1 (blue dotted), − 0.3 (red), 0.5 (green), and 1 (black dashed) are given in Figures 1(c) and 1(d), respectively
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