Rational Be´zier Line-Symmetric Motions

2005 ◽  
Vol 127 (2) ◽  
pp. 222-226 ◽  
Author(s):  
Shutian Li ◽  
Q. J. Ge
Keyword(s):  
Ruled Surface ◽  
Geometric Design ◽  
Ruled Surfaces ◽  
Line Geometry ◽  
Symmetric Motion ◽  
Computer Aided Geometric Design ◽  
De Casteljau Algorithm ◽  
Computer Aided ◽  
Kinematic Geometry

This paper brings together line geometry, kinematic geometry of line-symmetric motions, and computer aided geometric design to develop a method for geometric design of rational Be´zier line-symmetric motions. By taking advantage of the kinematic geometry of a line-symmetric motion, the problem of synthesizing a rational Be´zier line-symmetric motion is reduced to that of designing a rational Be´zier ruled surface. In this way, a recently developed de Casteljau algorithm for line-geometric design of ruled surfaces can be applied. An example is presented in which the Bennet motion is represented as a rational Be´zier line-symmetric motion whose basic surface is a hyperboloid.

Rational Bézier Line-Symmetric Motions

1999 ◽  
Author(s):  
Shutian Li ◽  
Q. J. Ge
Keyword(s):  
Ruled Surface ◽  
Geometric Design ◽  
Ruled Surfaces ◽  
Line Geometry ◽  
Symmetric Motion ◽  
Computer Aided Geometric Design ◽  
De Casteljau Algorithm ◽  
Computer Aided ◽  
Kinematic Geometry

Abstract This paper brings together line geometry, kinematic geometry of line-symmetric motions, and computer aided geometric design to develop a method for geometric design of rational Bézier line-symmetric motions. By taking advantage of the kinematic geometry of a line-symmetric motion, the problem of synthesizing a rational Bézier line-symmetric motion is reduced to that of designing a rational Bézier ruled surface. In this way, a recently developed de Casteljau algorithm for line-geometric design of ruled surfaces can be applied. An example is presented in which the Bennet motion is represented as a rational Bézier line-symmetric motion whose basic surface is a hyperboloid.

On Representation and Interpolation of Line-Segments for Computer Aided Geometric Design

1994 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Line Segment ◽  
Geometric Design ◽  
Ruled Surfaces ◽  
Line Geometry ◽  
Computer Aided Geometric Design ◽  
Line Segments ◽  
Geometric Shapes ◽  
Computer Aided ◽  
On Line ◽  
New Algorithms

Abstract In this paper, three new representations of a line-segment are introduced that combine Plücker line coordinates with specifications of length and location of a line-segment. For each of the three line-segment representations, a right conoidal interpolant of two arbitrarily disposed line-segments is developed. These interpolants are then combined with deCasteljau-like algorithms for generating ruled surfaces. The results can be extended to develop new algorithms for computer aided geometric design of geometric shapes based on line geometry.

scholarly journals Motion of Bishop Frenet Offsets of Ruled Surfaces inE3

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
H. N. Abd-Ellah
Keyword(s):  
Geometric Design ◽  
Ruled Surfaces ◽  
Computer Aided Geometric Design ◽  
Computer Aided

The main goal of this paper is to study the motion of two associated ruled surfaces in Euclidean 3-spaceE3. In particular, the motion of Bishop Frenet offsets of ruled surfaces is investigated. Additionally, the characteristic properties for such ruled surfaces are given. Finally, an application is presented and plotted using computer aided geometric design.

scholarly journals A new approach to design the ruled surface

2019 ◽  
Vol 16 (06) ◽  
pp. 1950093
Author(s):  
Ferhat Taş ◽  
Kazım İlarslan
Keyword(s):  
Unit Sphere ◽  
Ruled Surface ◽  
Geometric Design ◽  
Control Points ◽  
Computer Aided Geometric Design ◽  
New Approach ◽  
Transference Principle ◽  
Integral Invariants ◽  
Computer Aided ◽  
Novel Method

This paper considers a kind of design of a ruled surface. The design interconnects some concepts from the fields of computer-aided geometric design (CAGD) and kinematics. Dual unit spherical Bézier-like curves on the dual unit sphere (DUS) are obtained by a novel method with respect to the control points. A dual unit spherical Bézier-like curve corresponds to a ruled surface by using Study’s transference principle and closed ruled surfaces are determined via control points and also, integral invariants of these surfaces are investigated. Finally, the results are illustrated by several examples and the motion interpolation was shown as an embodiment of this method.

Sweeping surfaces according to type-2 Bishop frame in Euclidean 3-space

2021 ◽  
pp. 2150184
Author(s):  
Rashad A. Abdel-Baky ◽  
Fatemah Mofarreh
Keyword(s):  
Plane Curve ◽  
Special Kind ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Bishop Frame ◽  
Computer Aided ◽  
Kinematic Geometry

This work is concerned with the study of the kinematic-geometry of a special kind of tube surfaces, so-called sweeping surface in Euclidean 3-space [Formula: see text]. It is generated by a plane curve moving through space such that the movement of any point on the surface is always orthogonal to the plane. In particular, the type-2 Bishop frame is considered and some important theorems are obtained. Also, the problem of singularity and convexity of such sweeping surface is discussed. Finally, an application is presented and plotted using computer aided geometric design.

scholarly journals Compensated Evaluation of Tensor Product Surfaces in CAGD

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2219
Author(s):  
Jorge Delgado Gracia
Keyword(s):  
Error Analysis ◽  
Tensor Product ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
De Casteljau Algorithm ◽  
Usual Form ◽  
Computer Aided ◽  
Forward Error ◽  
Polynomial Surface

In computer-aided geometric design, a polynomial surface is usually represented in Bézier form. The usual form of evaluating such a surface is by using an extension of the de Casteljau algorithm. Using error-free transformations, a compensated version of this algorithm is presented, which improves the usual algorithm in terms of accuracy. A forward error analysis illustrating this fact is developed.

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.

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