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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.

The modeling of free-form engineering complex curves is an important subject in product modeling, graphics, and computer aided design/computer aided manufacturing (CAD/CAM). In this paper, we propose a novel method to construct free-form complex curves using shape-adjustable generalized Bézier (or SG-Bézier, for short) curves with constraints of geometric continuities. In order to overcome the difficulty that most of the composite curves in engineering cannot often be constructed by using only a single curve, we propose the necessary and sufficient conditions for G1 and G2 continuity between two adjacent SG-Bézier curves. Furthermore, the detailed steps of smooth continuity for two SG-Bézier curves, and the influence rules of shape parameters on the composite curves, are studied. We also give some important applications of SG-Bézier curves. The modeling examples show that our methods in this paper are very effective, can easily be performed, and can provide an alternative powerful strategy for the design of complex curves.

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In this paper, we propose a novel method for constructing developable surfaces using generalized C-Bézier bases with shape parameters. Based on the duality between points and planes in 3D projective space, the generalized developable C-Bézier surfaces, whose shape can be adjusted by changing multiple shape parameters, are designed using control planes with extensional C-Bézier basis functions. With the shape parameters taking different values, a family of developable surfaces can be constructed, which keeps most of characteristics of classic developable Bézier surfaces. Furthermore, some interesting properties of the new developable surfaces, as well as the geometric continuity conditions between two adjacent generalized developable C-Bézier surfaces, are investigated. Finally, we illustrate the convenience and efficiency of the proposed methods by several convictive and representative numerical examples.

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of and between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design.