scholarly journals Method and Code for the Visualization of Multivariate Solids

2000 ◽  
Author(s):  
Karim Abdel-Malek ◽  
Jingzhou Yang
Keyword(s):  
Computational Complexity ◽  
Implicit Function Theorem ◽  
Computer Code ◽  
Boundary Representation ◽  
Implicit Function ◽  
Computer Aided Geometric Design ◽  
Exact Boundary ◽  
Computer Aided ◽  
Geometric Entity ◽  
Swept Volume

Abstract This paper is devoted to a method and computer code for the automatic visualization of multivariate solids. Example of a multivariate solids arise in computer aided geometric design when a geometric entity is swept in space, where the totality of points touched by the entity is called the swept volume and is characterized by an equation of many parameters. The method and code are presented in an integrated manner and are aimed at providing the reader with a replicable computer algorithm. The formulation for is based on the implicit function theorem; is applicable to the visualization of solids of any number of parameters; and produces the exact boundary representation. Considering the solid as a manifold (possibly with boundaries), it is shown that further stratification of the various submanifolds yields varieties that can be depicted in R3. A measure of the computational complexity is presented to give the reader a sense of robustness of the method. The code is developed using a symbolic manipulator and is presented with a number of examples.

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.

Conformal Optimization Algorithm for Undevelopable Surfaces and its Application

2014 ◽  
Vol 602-605 ◽  
pp. 3570-3574
Author(s):  
Zhen Hua Luo ◽  
Fen Jiang
Keyword(s):  
Approximation Algorithms ◽  
Optimization Algorithm ◽  
Manufacturing Process ◽  
Research Topic ◽  
Important Research ◽  
Computer Aided Geometric Design ◽  
Industrial Manufacturing ◽  
Important Research Topic ◽  
Computer Aided ◽  
Planar Materials

In the industrial manufacturing process, most kinds of surfaces are processed by planar materials, but undevelopable surfaces are difficult develop to the plane. The approximation algorithms to develop a undevelopable surface is an important research topic in Computer Aided Geometric Design (CAGD). In this paper, we propose a new approximation algorithms based optimization algorithm. We guarantee the deformation vector make the minimum changes during the developing process. In the paper, some numerical example are given and the can illustrate the our method is effective.

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.

The Origin and Bifurcation of Disclinations in the Gauge Field Theory of Dislocation and Disclination Continuum

1998 ◽  
Vol 12 (25) ◽  
pp. 2599-2617 ◽  
Author(s):  
Guo-Hong Yang ◽  
Yishi Duan
Keyword(s):  
Field Theory ◽  
Gauge Field ◽  
Implicit Function Theorem ◽  
Taylor Expansion ◽  
Gauge Field Theory ◽  
Implicit Function ◽  
Quantum Numbers ◽  
Topological Quantum ◽  
Topological Current ◽  
Limit Points

In the 4-dimensional gauge field theory of dislocation and disclination continuum, the topological current structure and the topological quantization of disclinations are approached. Using the implicit function theorem and Taylor expansion, the origin and bifurcation theories of disclinations are detailed in the neighborhoods of limit points and bifurcation points, respectively. The branch solutions at the limit points and the different directions of all branch curves at 1-order and 2-order degenerated points are calculated. It is pointed out that an original disclination point can split into four disclinations at one time at most. Since the disclination current is identically conserved, the total topological quantum numbers of these branched disclinations will remain constant during their origin and bifurcation processes. Furthermore, one can see the fact that the origin and bifurcation of disclinations are not gradual changes but sudden changes. As some applications of the proposal theory, two examples are presented in the paper.

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