Abstract
The local controlled generalized H-Bézier model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is owed to its good geometric properties, e.g., symmetry and shape adjustable property. In this paper, some geometric continuity conditions for the generalized cubic H-Bézier model are studied for the purpose of constructing shape-controlled complex curves and surfaces in engineering. Firstly, based on the linear independence of generalized H-Bézier basis functions (GHBF), the conditions of first-order and second-order geometric continuity (namely, G1 and G2 continuity) between two adjacent generalized cubic H-Bézier curves are proposed. Furthermore, following analysis of the terminal properties of GHBF, the conditions of G1 geometric continuity between two adjacent generalized H-Bézier surfaces are derived and then simplified by choosing appropriate shape parameters. Finally, two operable procedures of smooth continuity for the generalized H-Bézier model are devised. Modeling examples show that the smooth continuity technology of the generalized H-Bézier model can improve the efficiency of computer design for complex curve and surface models.
Highlights
Parametric curves and surfaces are the main tools used to describe the geometric shape of products in computer-aided geometric design (CAGD), and their related theories and technologies are the bases and core contents of the whole CAGD [1,2]
We investigated some properties of the generalized H-Bézier model and derived the geometric conditions for G1 and G2 smooth continuity between two adjacent generalized H-Bézier curves and for G1 smooth continuity between two adjacent generalized H-Bézier surfaces
We developed two operable procedures of smooth continuity for generalized H-Bézier curves and surfaces, and we utilized several representative and convincing examples to verify the effectiveness of the proposed geometric continuity conditions
Summary
Parametric curves and surfaces are the main tools used to describe the geometric shape of products in computer-aided geometric design (CAGD), and their related theories and technologies are the bases and core contents of the whole CAGD [1,2]. I=0 where α is the shape parameter, P0 , P1 , P2 , P3 are the control points of cubic H-Bézier curves, and zi, (t) (i = 0, 1, 2, 3) are the basis functions of the cubic H-Bézier curves, defined as follows:. Wang [10] extended the definition of cubic H-Bézier curves to nth-degree curves They constructed a n o normative basis set in algebraic hyperbolic hybrid space Γn = span 1, t, · · · , tn−2 , sin ht, cos ht (n ≥ 2). H-Bézier model, in this paper we propose the respective geometric continuity conditions between two adjacent generalized H-Bézier curves and surfaces.
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