scholarly journals THE SHAPE OPERATOR OF THE BEZIER SURFACES IN MINKOWSKI-3 SPACE

2020 ◽  
Vol 20 (4) ◽  
pp. 865-880
Author(s):  
HATİCE KUŞAK SAMANCI ◽  
SERKAN ÇELİK ◽  
MUHSİN İNCESU
Keyword(s):  
Geometric Modeling ◽  
Shape Operator ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Bézier Surfaces ◽  
Computer Aided ◽  
Bezier Surfaces ◽  
Spacelike Surfaces

Bezier surfaces are commonly used in Computer-Aided Geometric Design since it enables in geometric modeling of the objects. In this study, the shape operator of the timelike and spacelike surfaces has been analyzed in Minkowski-3 space. Then, the obtained results were applied to a numeric example

scholarly journals Representation of the Matrix for Conversion between Triangular Bezier Patches and Rectangular Bezier Patches

2018 ◽  
Vol 22 ◽  
pp. 01020
Author(s):  
Pembe Sabancıgil ◽  
Mustafa Kara
Keyword(s):  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Simple Representation ◽  
Bézier Surfaces ◽  
Computer Aided ◽  
The Matrix ◽  
Conversion Matrix ◽  
One Step ◽  
Bezier Surfaces

In this paper we studied Bezier surfaces that are very famous techniques and they are widely used in the area of Computer Aided Geometric Design. Mainly there are two kinds of Bezier surfaces which are classified as rectangular and triangular Bezier patches. In this paper we will give a simple representation for the conversion matrix which converts one type to another type in one step.

scholarly journals Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

2021 ◽  
Vol 2 (5) ◽  
Author(s):  
Soroosh Tayebi Arasteh ◽  
Adam Kalisz
Keyword(s):  
Computer Graphics ◽  
Geometric Design ◽  
The Other ◽  
Control Points ◽  
Complex Surfaces ◽  
Linear Transformations ◽  
Computer Aided Geometric Design ◽  
Cubic Curves ◽  
Original Curve ◽  
Computer Aided

AbstractSplines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

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.

Kinematically Generated Dual Tensor-Product Surfaces

1998 ◽  
Author(s):  
Q. J. Ge ◽  
D. Kang ◽  
M. Sirchia
Keyword(s):  
Tensor Product ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Freeform Surfaces ◽  
Computer Aided ◽  
Design And Manufacture ◽  
Two Parameter

Abstract This paper takes advantage of the duality between point and plane geometries and studies a class of tensor-product surfaces that can be generated kinematically as surfaces enveloped by a plane under two-parameter rational Bézier motions. The results of this cross-disciplinary work, between the field of Computer Aided Geometric Design and Kinematics, can be used as a basis for studying geometric and kinematic issues associated with the design and manufacture of freeform surfaces.

Computer Aided Geometric Design of Two-Parameter Freeform Motions

1999 ◽  
Vol 121 (4) ◽  
pp. 502-506 ◽  
Author(s):  
Q. J. Ge ◽  
M. Sirchia
Keyword(s):  
Computer Aided Design ◽  
Control Structure ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Kinematic Control ◽  
Computer Aided ◽  
Cad Cam ◽  
Aided Design ◽  
Two Parameter ◽  
And Robotics

This paper brings together the notion of analytically defined two-parameter motion in Theoretical Kinematics and the notion of freeform surfaces in Computer Aided Geometric Design (CAGD) to develop methods for computer aided design of two-parameter freeform motions. In particular, a rational Be´zier representation for two-parameter freeform motions is developed. It has been shown that the trajectory surface of such a motion is a tensor-product rational Be´zier surface and that such a kinematically generated surface has a geometric as well as a kinematic control structure. The results have not only theoretical interest in CAGD and kinematics but also applications in CAD/CAM and Robotics.

A Simple Practical Criterion to Guarantee Second Order Smoothness of Blend Surfaces

1989 ◽  
Author(s):  
J. Pegna ◽  
F.-E. Wolter
Keyword(s):  
Second Order ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Common Boundary ◽  
First Order ◽  
Computer Aided ◽  
The Common ◽  
Blend Surfaces ◽  
Continuous Curves ◽  
Order Smoothness

Abstract Computer Aided Geometric Design of surfaces sometimes presents problems that were not envisioned by mathematicians in differential geometry. This paper presents mathematical results that pertain to the design of second order smooth blending surfaces. Second order smoothness normally requires that normal curvatures agree along all tangent directions at all points of the common boundary of two patches, called the linkage curve. The Linkage Curve Theorem proved here shows that, for the blend to be second order smooth when it is already first order smooth, it is sufficient that normal curvatures agree in one direction other than the tangent to a first order continuous linkage curve. This result is significant for it substantiates earlier works in computer aided geometric design. It also offers simple practical means of generating second order blends for it reduces the dimensionality of the problem to that of curve fairing, and is well adapted to a formulation of the blend surface using sweeps. From a theoretical viewpoint, it is remarkable that one can generate second order smooth blends with the assumption that the linkage curve is only first order smooth. This property may be helpful to the designer since linkage curves can be constructed from low order piecewise continuous curves.

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