scholarly journals Generalized Developable Cubic Trigonometric Bézier Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 283
Author(s):  
Muhammad Ammad ◽  
Md Yushalify Misro ◽  
Muhammad Abbas ◽  
Abdul Majeed
Keyword(s):  
Shape Parameter ◽  
Surface Characteristics ◽  
Surface Shape ◽  
Shape Parameters ◽  
New Approach ◽  
Bézier Surface ◽  
Bezier Surface ◽  
Bézier Surfaces ◽  
Bezier Surfaces ◽  
Shape Adjustment

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.

scholarly journals Bezier Surfaces and Texture Mapping Using Java 3D

2012 ◽  
Vol 6-7 ◽  
pp. 1000-1003
Author(s):  
Xin Rui Gao
Keyword(s):  
Texture Mapping ◽  
Surface Model ◽  
Control Points ◽  
Java 3D ◽  
Mapping Algorithms ◽  
Bézier Surface ◽  
Bezier Surface ◽  
Bézier Surfaces ◽  
Matrix Formula ◽  
Bezier Surfaces

By using Bezier surface matrix formula and the algorithms of texture mapping, the texture mapping onto Bezier surface and its control points net were tested. The texture mapping for Bezier surface model that is composed of six Bezier surfaces was tested too. From the testing examples, it is concluded that these texture mapping algorithms are reliable. All algorithms were implemented by Java and Java 3D.

Three Order Bezier Surface and Racing Car Conception Model Design Using Java 3D

2011 ◽  
Vol 58-60 ◽  
pp. 1272-1276
Author(s):  
Xin Rui Gao ◽  
Yong Chuan Zhang
Keyword(s):  
Control Points ◽  
Model Design ◽  
Java 3D ◽  
Bézier Surface ◽  
Car Model ◽  
Bezier Surface ◽  
Surface Control ◽  
Bézier Surfaces ◽  
Matrix Formula ◽  
Bezier Surfaces

By using Bezier surface matrix formula and Bezier surface control points and through merging different Bezier surfaces, the car model was produced. By using texture and transparence and other functions of Java 3D, the tyres and the wind screen of the car and the exhibition platform were designed. By defining the moving and rotation actions, the motions of the car model were designed. All functions were implemented by Java and Java 3D.

scholarly journals Variationally Improved Bézier Surfaces with Shifted Knots

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Daud Ahmad ◽  
Kanwal Hassan ◽  
M. Khalid Mahmood ◽  
Javaid Ali ◽  
Ilyas Khan ◽  
...  
Keyword(s):  
Control Point ◽  
Linear Constraints ◽  
Control Points ◽  
Bézier Surface ◽  
Bezier Surface ◽  
Bézier Surfaces ◽  
Variational Minimization ◽  
Area Functional ◽  
Bezier Surfaces ◽  
Minimal Area

The Plateau-Bézier problem with shifted knots is to find the surface of minimal area amongst all the Bézier surfaces with shifted knots spanned by the admitted boundary. Instead of variational minimization of usual area functional, the quasi-minimal Bézier surface with shifted knots is obtained as the solution of variational minimization of Dirichlet functional that turns up as the sum of two integrals and the vanishing condition gives us the system of linear algebraic constraints on the control points. The coefficients of these control points bear symmetry for the pair of summation indices as well as for the pair of free indices. These linear constraints are then solved for unknown interior control points in terms of given boundary control points to get quasi-minimal Bézier surface with shifted knots. The functional gradient of the surface gives possible candidate functions as the minimizers of the aforementioned Dirichlet functional; when solved for unknown interior control points, it results in a surface of minimal area called quasi-minimal Bézier surface. In particular, it is implemented on a biquadratic Bézier surface by expressing the unknown control point P 11 as the linear combination of the known control points in this case. This can be implemented to Bézier surfaces with shifted knots of higher degree, as well if desired.

Construction of generalized developable Bézier surfaces with shape parameters

2018 ◽  
Vol 41 (17) ◽  
pp. 7804-7829 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Xinqiang Qin
Keyword(s):  
Shape Parameters ◽  
Bézier Surfaces ◽  
Bezier Surfaces

scholarly journals G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2350
Author(s):  
Samia BiBi ◽  
Md Yushalify Misro ◽  
Muhammad Abbas ◽  
Abdul Majeed ◽  
Tahir Nazir
Keyword(s):  
Shape Control ◽  
Duality Principle ◽  
Developable Surface ◽  
Control Parameters ◽  
Shape Parameters ◽  
Developable Surfaces ◽  
Bézier Surfaces ◽  
Canal Surfaces ◽  
Novel Method ◽  
Bezier Surfaces

In this article, we proposed a novel method for the construction of generalized hybrid trigonometric (GHT-Bézier) developable surfaces to tackle the issue of modeling and shape designing in engineering. The GHT-Bézier developable surface is obtained by using the duality principle between the points and planes with GHT-Bézier curve. With different shape control parameters in their domain, a class of GHT-Bézier developable surfaces can be established (such as enveloping developable GHT-Bézier surfaces, spine curve developable GHT-Bézier surfaces, geodesic interpolating surfaces for GHT-Bézier surface and developable GHT-Bézier canal surfaces), which possess many properties of GHT-Bézier surfaces. By changing the values of shape parameters the effect on the developable surface is obvious. In addition, some useful geometric properties of GHT-Bézier developable surface and the G1, G2 (Farin-Boehm and Beta) and G3 continuity conditions between any two GHT-Bézier developable surfaces are derived. Furthermore, various useful and representative numerical examples demonstrate the convenience and efficiency of the proposed method.

scholarly journals Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Abdul Majeed ◽  
Gang Hu ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Architectural Design ◽  
Duality Principle ◽  
Shape Parameters ◽  
Developable Surfaces ◽  
Bézier Surfaces ◽  
Current Curve ◽  
Bezier Surfaces ◽  
Necessary And Sufficient

AbstractDevelopable 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 $G^{1}$ G 1 and $G^{2}$ 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.

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