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.

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.

Design for Manufacturing Using B-Spline Developable Surfaces

1998 ◽  
Vol 42 (03) ◽  
pp. 207-215 ◽  
Author(s):  
Julie S. Chalfant ◽  
Takashi Maekawa
Keyword(s):  
Heat Treatment ◽  
Design For Manufacturing ◽  
Optimization Techniques ◽  
Developable Surface ◽  
Planar Surface ◽  
Complex Objects ◽  
B Spline ◽  
Developable Surfaces ◽  
Manufacturing Applications ◽  
User Friendly

A developable surface can be formed by bending or rolling a planar surface without stretching or tearing; in other words, it can be developed or unrolled isometrically onto a plane. Developable surfaces are widely used in the manufacture of items that use materials that are not amenable to stretching such as the formation of ducts, shoes, clothing and automobile parts including upholstery and body panels (Frey & Bindschadler 1993). Designing a ship hull entirely of developable surfaces would allow production of the hull using only rolling or bending. Heat treatment would only be required for removal of distortion, thus greatly reducing the labor required to form the hull. Although developable surfaces play an important role in various manufacturing applications, little attention has been paid to implementing developable surfaces from the onset of a design. This paper investigates novel, user friendly methods to design complex objects using B-spline developable surfaces based on optimization techniques. Illustrative examples show the substantial improvements this method achieves over previously developed methods.

Approximate Merging B-Spline Curves via Least Square Approximation

2014 ◽  
Vol 556-562 ◽  
pp. 3496-3500 ◽  
Author(s):  
Si Hui Shu ◽  
Zi Zhi Lin
Keyword(s):  
Approximation Error ◽  
Least Square ◽  
Error Tolerance ◽  
Knot Insertion ◽  
Least Squares Approximation ◽  
B Spline ◽  
Spline Curve ◽  
Spline Curves ◽  
Insertion Algorithm ◽  
Approximate Merging

An algorithm of B-spline curve approximate merging of two adjacent B-spline curves is presented in this paper. In this algorithm, the approximation error between two curves is computed using norm which is known as best least square approximation. We develop a method based on weighed and constrained least squares approximation, which adds a weight function in object function to reduce error of merging. The knot insertion algorithm is also developed to meet the error tolerance.

Another knot insertion algorithm for B-spline curves

1992 ◽  
Vol 9 (3) ◽  
pp. 175-183 ◽  
Author(s):  
Phillip J. Barry ◽  
Rui-Feng Zhu
Keyword(s):  
Knot Insertion ◽  
B Spline ◽  
Spline Curves ◽  
Insertion Algorithm

scholarly journals A Tidal Level Prediction Approach Based on BP Neural Network and Cubic B-Spline Curve with Knot Insertion Algorithm

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Wenjuan Wang ◽  
Hongchun Yuan
Keyword(s):  
Neural Network ◽  
Storm Surge ◽  
Bp Neural Network ◽  
Knot Insertion ◽  
Tidal Level ◽  
B Spline ◽  
Spline Curve ◽  
Tide Level ◽  
Insertion Algorithm ◽  
Prediction Approach

Tide levels depend on both long-term astronomical effects that are mainly affected by moon and sun and short-term meteorological effects generated by severe weather conditions like storm surge. Storm surge caused by typhoons will impose serious security risks and threats on the coastal residents’ safety in production, property, and life. Due to the challenges of nonperiodic and incontinuous tidal level record data and the influence of multimeteorological factors, the existing methods cannot predict the tide levels affected by typhoons precisely. This paper targets to explore a more advanced method for forecasting the tide levels of storm surge caused by typhoons. First, on the basis of successive five-year tide level and typhoon data at Luchaogang, China, a BP neural network model is developed using six parameters of typhoons as input parameters and the relevant tide level data as output parameters. Then, for an improved forecasting accuracy, cubic B-spline curve with knot insertion algorithm is combined with the BP model to conduct smooth processing of the predicted points and thus the smoothed prediction curve of tidal level has been obtained. By using the data of the fifth year as the testing sample, the predicted results by the two methods are compared. The experimental results have shown that the latter approach has higher accuracy in forecasting tidal level of storm surge caused by typhoons, and the combined prediction approach provides a powerful tool for defending and reducing storm surge disaster.

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.

Geometric Design of Uniform Developable B-Spline Surfaces

2004 ◽  
Author(s):  
Chih-Hsing Chu ◽  
Jang-Ting Chen
Keyword(s):  
Degrees Of Freedom ◽  
Solution Process ◽  
Boundary Curve ◽  
Geometric Design ◽  
Constrained Systems ◽  
Control Points ◽  
B Spline ◽  
Cubic Surface ◽  
De Boor Algorithm ◽  
Spline Surfaces

This paper studies geometric design of uniform developable B-spline surfaces from two boundary curves. The developability constraints are geometrically derived from the de Boor algorithm and expressed as a set of equations that must be fulfilled by the B-spline control points. These equations help characterize the number of degrees of freedom (DOF’s) for the surface design. For a cubic B-spline surface with a first boundary curve freely chosen, five more DOF’s are available for a second boundary curve when both curves contain four control points. There remain (7-2m) DOF’s for a cubic surface consisting of m consecutive patches with C2 continuity. The results are in accordance with previous findings for equivalent composite Be´zier surfaces. Test examples are illustrated to demonstrate design methods that fully utilize the DOF’s without leading to over-constrained systems in the solution process. Providing a foundation for systematic implementation of a CAGD system for developable B-spline surfaces, this work has substantial improvements over past studies.

BLIND WATERMARKING OF NON-UNIFORM B-SPLINE SURFACES

2008 ◽  
Vol 08 (03) ◽  
pp. 439-454 ◽  
Author(s):  
SHUSEN SUN ◽  
ZHIGENG PAN ◽  
TAE-WAN KIM
Keyword(s):  
Blind Watermarking ◽  
Knot Insertion ◽  
B Spline ◽  
Original Surface ◽  
Spline Surfaces ◽  
Sign Correlation ◽  
Dct Coefficients ◽  
Sample Points ◽  
Watermarking Scheme ◽  
Correlation Detector

In this paper, we propose a watermarking scheme for non-uniform B-spline (NUBS) surface. Firstly, we first do sampling on a NUBS surface and get the sample points, then the watermark is embedded into the DCT coefficients of the sample points and the watermarked sample points are transformed back, finally the watermarked surface is reconstructed from watermarked sample points using global interpolation. A sign correlation detector is used to test for the presence of the watermark, and the original surface is not required at this stage. Experimental results show that our algorithm can preserve the shape of the original surface within a specified error, and that it is robust against attacks including knot insertion, order elevation, addition of white noise, rotation, scaling, translation and further watermarking.

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