G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications
Abstract
:1. Introduction
- The representation of a family of GHT-Bernstein basis functions and GHT-Bézier curves with geometric properties.
- The designing of Enveloping developable GHT-Bézier surfaces, surfaces interpolation geodesic developable GHT-Bézier curve with shape control parameters, Spine curve developable GHT-Bézier surfaces, and developable GHT-Bézier Canal surfaces based on computer technology.
- continuity and continuity of developable GHT-Bézier surfaces are also presented. Moreover, an algorithm is given to explain how to enforce these continuities in practice.
- This approach becomes more convenient because of the exclusion of difficult calculations which come from the nonlinearity of characteristic equations.
2. A Family of GHT-Bernstein Basis with Geometric Properties
3. GHT-Bézier Curves with Geometric Properties
4. Composition of GHT-Bézier Developable Surfaces
4.1. Dual Representation of One-Parameter Family of Planes
4.2. Enveloping Developable GHT-Bézier Surfaces
4.3. Spine Curve Developable GHT-Bézier Surfaces
4.4. Developable Surface Interpolating Geodesic GHT-Bézier Curve
4.5. Developable GHT-Bézier Canal Surface
5. Analysis Properties of Developable GHT-Bézier Surfaces
6. Continuity Conditions of Developable GHT-Bézier Surfaces
6.1. Axioms of Continuity between Developable GHT-Bézier Surfaces
6.2. Axioms of Farin-Boehm Continuity between Developable GHT-Bézier Surfaces
6.3. Axioms of Beta Continuity between Developable GHT-Bézier Surfaces
6.4. Axioms of Continuity between Developable GHT-Bézier Surfaces
6.5. Algorithm to Explain Axioms of Continuity for Developable GHT-Bézier Surfaces
- Consider the shape parameters and control planes of initial developable GHT-Bézier surface
- For continuity, the developable GHT-Bézier surfaces and have a common control plane.
- The second control plane of developable GHT-Bézier surface can be computed by using the second condition of Equation (31). Here the values of shape parameters, chosen freely from their domain while scale factors, can be any real constant.
- For continuity, the third control plane of surface can be obtained by using Farin-Boehm and Beta continuity condition between two adjacent developable GHT-Bézier surfaces.
- The last control plane of developable GHT-Bézier surface can be obtained by using the last condition of Equation (35), i.e., the continuity between two adjacent developable GHT-Bézier surfaces.
7. Designing of Developable GHT-Bézier Surfaces
7.1. Examples for the Construction of Enveloping Developable GHT-Bézier Surfaces
- When all the shape parameters varies equally, then length of starting generator and ending generator will remain the same, but at the same time the enveloping developable surface as shown in Figure 3 will expand (or shrink).
- When two shape parameters are fixed to 1 and varies, then length of starting generator and ending generator are closely related (are same) and enveloping developable GHT-Bézier surface will shrink.
- When shape parameters are fixed to 1 and varies, then length of starting generator is greater then the length of ending generator as shown in Figure 4d–f.
- When shape parameters are fixed to 1 and varies, then length of ending generator is greater then the length of starting generator and enveloping developable GHT-Bézier surface will expand (or shrink) gradually as shown in Figure 4g–i.
7.2. Spine Curve Developable GHT-Bézier Surface Construction
- When we consider all the shape parameters, varies equally with negative values, i.e., then length of starting generator increases with length of ending generator but when we have positive values of shape parameters, the length of starting generator decreases with length of ending generator. The behavior of shape parameters will shrink/expand, the shape of the surface as well.
- When two shape parameters are fixed to 0 and varies, then length of ending generator is larger then the length of starting generator However, as we increase the values of the surface will shrink and the length of starting generator will also increase.
- When are fixed to 0 and values of increase, then the length of starting generator decreases and length of ending generator increases. However, at the same time the spine curve developable GHT-Bézier surface will expand.
- When the shape parameters are fixed to and varies, then the length of starting generator increases and length of ending generator remains same. The surface will shrink by decreasing the values of shape parameters.
7.3. Designing of Developable Surface Interpolating Geodesic GHT-Bézier Curve with Parameters
- When shape control parameters vary simultaneously, then smooth behavior of surface interpolation geodesic developable surfaces is shown in Figure 7a–c.
- When we fix two shape parameters and vary the remaining shape parameters one by one, then the behavior of developable surfaces also varies and is obvious in Figure 7d–i. The values of shape parameters are mentioned under each figure.
7.4. Construction of Developable GHT-Bézier Canal Surface with Shape Parameters
7.5. Examples of Smooth Continuity between Two Adjacent Developable GHT-Bézier Surfaces
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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BiBi, S.; Misro, M.Y.; Abbas, M.; Majeed, A.; Nazir, T. G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications. Mathematics 2021, 9, 2350. https://doi.org/10.3390/math9192350
BiBi S, Misro MY, Abbas M, Majeed A, Nazir T. G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications. Mathematics. 2021; 9(19):2350. https://doi.org/10.3390/math9192350
Chicago/Turabian StyleBiBi, Samia, Md Yushalify Misro, Muhammad Abbas, Abdul Majeed, and Tahir Nazir. 2021. "G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications" Mathematics 9, no. 19: 2350. https://doi.org/10.3390/math9192350
APA StyleBiBi, S., Misro, M. Y., Abbas, M., Majeed, A., & Nazir, T. (2021). G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications. Mathematics, 9(19), 2350. https://doi.org/10.3390/math9192350