Abstract
Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G^{1} and G^{2} (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.
Highlights
Due to the ease of engineering procedure, developable surfaces are especially fascinating and tempting
Actual developable surfaces have an instinctive implementation in several fields of engineering and manufacturing as an aircraft architect uses them to form airplane wings and a tinsmith uses them to attach two tubes of different designs with flattened sections of metal sheets
With different values of shape parameters μ and ν, a class of cubic spine curve developable GBT-Bézier surfaces can be designed on the requirement of provided control planes Q0, Q1, Q2, Q3 in Fig. 13 and Fig. 14 respectively, with distinct shapes
Summary
Due to the ease of engineering procedure, developable surfaces are especially fascinating and tempting. The research issue for the construction and designing of developable surfaces is consistently important in CAD/CAM [2, 3] as it is concerned with modeling and invigorating objects which are examined in daily life In this context, Chung et al [1] suggested a technique to make shoe uppers by taking triangles and to improve the surface to make it more developable. Algorithms that present the developable surfaces using Bézier curve of arbitrary shape and order were generated by Aumann [5, 6] in which the control of singular points is insured. Li, Hu et al, and Ammad et al proposed the designing approaches for developable C-Bézier [27], cubic developable C-Bézier surfaces [28], and generalized developable cubic trigonometric Bézier surfaces [29], respectively These schemes bring a beneficial opportunity for the developable surfaces to the actual geometric modeling techniques.
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