Parametric Continuity and Smooth Motion Interpolation

1995 ◽  
Author(s):  
Lakshmi N. Srinivasan ◽  
Q. J. Ge
Keyword(s):  
Mechanical Systems ◽  
Trajectory Generation ◽  
Geometric Design ◽  
Second Derivative ◽  
Computer Aided Geometric Design ◽  
Smooth Motion ◽  
Computer Aided ◽  
Cad Cam ◽  
C2 Continuity ◽  
Parametric Continuity

Abstract This paper deals with the design of a second derivative continuous (C2) motion that interpolates through a given set of configurations of an object. It derives conditions for blending two motion segments with C2 continuity and develops an algorithm for constructing a C2 composite Bézier type motion that has similarities to Beta-splines in the field of Computer Aided Geometric Design. A criteria for evaluating the smoothness of motion is established and is used to synthesize “globally smooth” motions. The results have applications in trajectory generation in robotics, mechanical systems animation and CAD/CAM.

Parametric Continuous and Smooth Motion Interpolation

1996 ◽  
Vol 118 (4) ◽  
pp. 494-498 ◽  
Author(s):  
L. N. Srinivasan ◽  
Q. J. Ge
Keyword(s):  
Mechanical Systems ◽  
Trajectory Generation ◽  
Second Order ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Smooth Motion ◽  
Computer Aided ◽  
Cad Cam ◽  
C2 Continuity

This paper deals with the synthesis of a second order parametrically continuous (C2) motion that interpolates through a given set of configurations of an object. It derives conditions for blending two motion segments with C2 continuity and develops an algorithm for constructing a C2 composite Be´zier type motion that has similarities to Beta-splines in the field of Computer Aided Geometric Design. A criterion for evaluating the smoothness of a motion is established and is used to synthesize a “globally smooth” motion. The results have applications in trajectory generation in robotics, mechanical systems animation and CAD/CAM.

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.

Computer Aided Geometric Design of Two-Parameter Freeform Motions

1998 ◽  
Author(s):  
Q. J. Ge ◽  
M. Sirchia
Keyword(s):  
Computer Aided Design ◽  
Control Structure ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Computer Aided ◽  
Cad Cam ◽  
Rational Bézier Surface ◽  
Aided Design ◽  
Two Parameter ◽  
And Robotics

Abstract 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 Bézier 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 Bézier 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.

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.

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