Abstract
This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for G1, Farin–Boehm G2, and G2 Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.
Highlights
The developable structure, as a kind of special and meaningful ruled structure, may be expanded onto a plane rather than being stretched or broken
This section involves the construction of developable surface through a given cubic trigonometric Bézier curve, where the curve is the geodesic of the surface
The first Equations of (17) and (18) shows that the first and last planes corresponding to u = 0 and u = 1 in {∏u } are given by the designer as the first and last control planes and they are both tangential to the generalized developable cubic trigonometric (GDCT)-Bézier surface along its generator, at u = 0 and u = 1
Summary
The developable structure, as a kind of special and meaningful ruled structure, may be expanded onto a plane rather than being stretched or broken. Since a developable surface conserves tangent planes along each of its rulings, it is transformed into a curve in the Laguerre geometry [20]. [26,27] constructed a QTB-spline and a rational cubic trigonometric (CT)-Bézier curve with adjusting parameters respectively. Majeed and Qayyum [28] constructed a new rational cubic trigonometric B-spline where this proposed curve applies with various applicability and flexibility by using different weights and shape parameters. This paper attempts to resolve the issue of shape handling in developing surfaces using the cubic trigonometric Bézier basis functions and to analyze their favorable properties by extending the discussion to curves and surfaces. By enveloping developable and tangent curves of the spine, we build developable surfaces using a generalized developable cubic trigonometric Bézier (GDCT-Bézier) basis function.
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