Algebraic Motion Approximation With NURBS Motions and Its Application to Spherical Mechanism Synthesis

1999 ◽  
Vol 121 (4) ◽  
pp. 529-532 ◽  
Author(s):  
Q. Jeffrey Ge ◽  
P. M. Larochelle
Keyword(s):  
Algebraic Curves ◽  
Approximation Problem ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
B Spline ◽  
Approximation Techniques ◽  
Spline Curves ◽  
Curve Approximation ◽  
Computer Aided ◽  
Motion Approximation

In this work we bring together classical mechanism theory with recent works in the area of Computer Aided Geometric Design (CAGD) of rational motions as well as curve approximation techniques in CAGD to study the problem of mechanism motion approximation from a computational geometric viewpoint. We present a framework for approximating algebraic motions of spherical mechanisms with rational B-Spline spherical motions. Algebraic spherical motions and rational B-spline spherical motions are represented as algebraic curves and rational B-Spline curves in the space of quaternions (or the image space). Thus the problem of motion approximation is transformed into a curve approximation problem, where concepts and techniques in the field of Computer Aided Geometric Design and Computational Geometry may be applied. An example is included at the end to show how a NURBS motion can be used for synthesizing spherical four-bar linkages.

Algebraic Motion Approximation With NURBS Motions and its Application to Spherical Mechanism Synthesis

1998 ◽  
Author(s):  
Q. Jeffrey Ge ◽  
Pierre M. Larochelle
Keyword(s):  
Algebraic Curves ◽  
Approximation Problem ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
B Spline ◽  
Approximation Techniques ◽  
Spline Curves ◽  
Curve Approximation ◽  
Computer Aided ◽  
Motion Approximation

Abstract In this work we bring together classical mechanism theory with recent works in the area of Computer Aided Geometric Design (CAGD) of rational motions as well as curve approximation techniques in CAGD to study the problem of mechanism motion approximation from a computational geometric viewpoint. We present a framework for approximating algebraic motions of spherical mechanisms with rational B-Spline spherical motions. Algebraic spherical motions and rational B-spline spherical motions are represented as algebraic curves and rational B-Spline curves in the space of quaternions (or the image space). Thus the problem of motion approximation is transformed into a curve approximation problem, where concepts and techniques in the field of Computer Aided Geometric Design and Computational Geometry may be applied. An example is included at the end to show how a NURBS motion can be used for synthesizing spherical four-bar linkages.

Multiresolution Curve Editing With Linear Constraints*

2001 ◽  
Vol 1 (4) ◽  
pp. 347-355 ◽  
Author(s):  
Gershon Elber
Keyword(s):  
Linear Constraints ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Freeform Surfaces ◽  
Modeling Tool ◽  
B Spline ◽  
Curves And Surfaces ◽  
Orthogonality Constraints ◽  
Spline Curves ◽  
Computer Aided

The use of multiresolution control toward the editing of freeform curves and surfaces has already been recognized as a valuable modeling tool [1–3]. Similarly, in contemporary computer aided geometric design, the use of constraints to precisely prescribe freeform shape is considered an essential capability [4,5]. This paper presents a scheme that combines multiresolution control with linear constraints into one framework, allowing one to perform multiresolution manipulation of nonuniform B-spline curves, while specifying and satisfying various linear constraints on the curves. Positional, tangential, and orthogonality constraints are all linear and can be easily incorporated into a multiresolution freeform curve editing environment, as will be shown. Moreover, we also show that the symmetry as well as the area constraints can be reformulated as linear constraints and similarly incorporated. The presented framework is extendible and we also portray this same framework in the context of freeform surfaces.

Computer Aided Geometric Design of Motion Interpolants

1994 ◽  
Vol 116 (3) ◽  
pp. 756-762 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Computer Animation ◽  
Three Dimensional ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Mapping Space ◽  
Three Dimensional Objects ◽  
Spline Curves ◽  
Computer Aided ◽  
Curvature And Torsion ◽  
Spatial Kinematics

This paper studies continuous computational geometry of motion and develops a method for Computer Aided Geometric Design (CAGD) of motion interpolants. The approach uses a mapping of spatial kinematics to convert the problem of interpolating displacements to that of interpolating points in the space of the mapping. To facilitate the point interpolation, the previously unorientable mapping space is made orientable. Methods are then developed for designing spline curves in the mapping space with tangent, curvature and torsion continuities. The results have application in computer animation of three-dimensional objects used in computer graphics, computer vision and simulation of mechanical systems.

On the Effects of Weights in Rational Motion Design

2003 ◽  
Author(s):  
A. Purwar ◽  
Q. J. Ge
Keyword(s):  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Surface Design ◽  
Computer Aided ◽  
Motion Approximation ◽  
Motion Design

The effect of weights on curves and surface design is a well-researched topic in Computer Aided Geometric Design (CAGD). However, the influence of weights in the realm of rational motion approximation and interpolation has been largely unexplored. In this paper, we present a thorough mathematical exposition on the influence of weights on rational motion design. This leads naturally to providing the motion designer with guidelines on how to use weights for rational motion design. Several examples are presented towards the end.

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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