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.

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.

G1 Continuity of C-Bezier Rectangular Surfaces

2000 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Control Point ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Design Parameters ◽  
Control Points ◽  
Bézier Surfaces ◽  
G1 Continuity ◽  
Bezier Surfaces ◽  
Necessary And Sufficient

Abstract This paper probes G1 continuity between two adjacent c-Bezier rectangular patches. The necessary and sufficient conditions are derived. It shows that the coplanar condition for G1 continuity of two adjacent Bezier patches is not necessary for c-Bezier patches. Such a relaxation of constraints on control points is beneficial from vector weights of c-Bezier surfaces, which leads to two extra free design parameters for each control point. C-Bezier surfaces offer the possibility of obtaining G1 continuity by just adjusting the weights, which greatly simplifies the design to construct composite surfaces.

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.

Blending surfaces generated using the Bernstein operator

2012 ◽  
Vol 21 (1) ◽  
pp. 35-40
Author(s):  
MARIUS BIROU ◽  
Keyword(s):  
Control Points ◽  
Bernstein Operator ◽  
Bézier Surfaces ◽  
Bezier Surfaces

In this paper we construct blending surfaces using the univariate Bernstein operator. The surfaces have the properties that they stay on a curve (the border of the surfaces domain) and have a fixed height in a point from the domain. The surfaces are generated using a curve network, instead of the control points from the case of classical Bezier surfaces. We study the monotonicity and we give conditions to obtain concave surfaces.

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