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.

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 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.

Geometric Design and Fabrication of Developable Bezier Surfaces

1992 ◽  
Author(s):  
R. M. C. Bodduluri ◽  
B. Ravani
Keyword(s):  
Geometric Design ◽  
Developable Surface ◽  
Developable Surfaces ◽  
Surface Patches ◽  
Bézier Surfaces ◽  
Point Representation ◽  
Cad Cam ◽  
C2 Continuity ◽  
Type Construction ◽  
Bezier Surfaces

Abstract In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop direct representations of developable surfaces in terms of point as well as plane geometries. The point representation uses a Bezier curve, the tangents of which span the surface. The plane representation uses control planes instead of control points and determines a surface which is a Bezier interpolation of the control planes. In this case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In design of piecewise surface patches, a computational geometric algorithm similar to Farin-Boehm construction used in design of piecewise parametric curves is developed for designing developable surfaces with C2 continuity. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

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