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.

Geometric Design and Fabrication of Developable Bezier and B-spline Surfaces

1994 ◽  
Vol 116 (4) ◽  
pp. 1042-1048 ◽  
Author(s):  
R. M. C. Bodduluri ◽  
B. Ravani
Keyword(s):  
Spline Interpolation ◽  
Geometric Design ◽  
Developable Surface ◽  
Knot Insertion ◽  
B Spline ◽  
Developable Surfaces ◽  
Spline Surfaces ◽  
Insertion Algorithm ◽  
Cad Cam ◽  
Type Construction

In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop a direct representation of developable surfaces in terms of plane geometry. It uses control planes to determine a surface which is a Bezier or a B-spline interpolation of the control planes. In the Bezier case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In the B-spline case, de Boor type construction for the geometric design of the developable surface and Boehm type knot insertion algorithm are presented. 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.

scholarly journals G3 Shape Adjustable GHT-Bézier Developable Surfaces and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2350
Author(s):  
Samia BiBi ◽  
Md Yushalify Misro ◽  
Muhammad Abbas ◽  
Abdul Majeed ◽  
Tahir Nazir
Keyword(s):  
Shape Control ◽  
Duality Principle ◽  
Developable Surface ◽  
Control Parameters ◽  
Shape Parameters ◽  
Developable Surfaces ◽  
Bézier Surfaces ◽  
Canal Surfaces ◽  
Novel Method ◽  
Bezier Surfaces

In this article, we proposed a novel method for the construction of generalized hybrid trigonometric (GHT-Bézier) developable surfaces to tackle the issue of modeling and shape designing in engineering. The GHT-Bézier developable surface is obtained by using the duality principle between the points and planes with GHT-Bézier curve. With different shape control parameters in their domain, a class of GHT-Bézier developable surfaces can be established (such as enveloping developable GHT-Bézier surfaces, spine curve developable GHT-Bézier surfaces, geodesic interpolating surfaces for GHT-Bézier surface and developable GHT-Bézier canal surfaces), which possess many properties of GHT-Bézier surfaces. By changing the values of shape parameters the effect on the developable surface is obvious. In addition, some useful geometric properties of GHT-Bézier developable surface and the G1, G2 (Farin-Boehm and Beta) and G3 continuity conditions between any two GHT-Bézier developable surfaces are derived. Furthermore, various useful and representative numerical examples demonstrate the convenience and efficiency of the proposed method.

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.

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 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

Laser Forming Oriented CAD/CAM for Developable Surfaces

2007 ◽  
Vol 344 ◽  
pp. 905-912
Author(s):  
B. Callebaut ◽  
Joost R. Duflou ◽  
Jean Pierre Kruth
Keyword(s):  
Sheet Material ◽  
Surface Damage ◽  
Free Form ◽  
Laser Forming ◽  
Parabolic Cylinder ◽  
Developable Surface ◽  
Ship Building ◽  
Developable Surfaces ◽  
Cad Cam ◽  
Zero Gaussian Curvature

Laser forming of sheet material has been widely investigated for the last 15 years. While researchers encounter severe problems during the forming of a 3D free form shape, at least one category of surfaces can be made with the process of laser forming, namely the developable surfaces, which are widely used in, for example, ship building. Those surfaces show a zero gaussian curvature and can be unrolled onto a plane without distortion. Until now, the forming of such surfaces has been more or less heuristic, but this paper aims to treat the CAD/CAM issues of this problem in a generic way. Once the surface has been defined, in order to obtain a developable surface, the surface is rebuilt into a number of planar flanges. After collision testing, the unfolding of the surface is calculated. The developable surface is scanned on the boundary between two flanges using laser settings that are determined based on efficiency optimisation considerations, keeping in mind the hardware limitations and the possible surface damage for a too high input energy. In this paper, the proposed CAD/CAM procedure is validated by means of a developable parabolic cylinder.

scholarly journals Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sidra Maqsood ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Abdul Majeed ◽  
Gang Hu ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Architectural Design ◽  
Duality Principle ◽  
Shape Parameters ◽  
Developable Surfaces ◽  
Bézier Surfaces ◽  
Current Curve ◽  
Bezier Surfaces ◽  
Necessary And Sufficient

AbstractDevelopable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient $G^{1}$ G 1 and $G^{2}$ G 2 (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.

Sign in / Sign up

Export Citation Format

Share Document