scholarly journals Construction of Developable Surface with Geodesic or Line of Curvature Coordinates

2021 ◽  
pp. 75-87
Author(s):  
Nabil ALTHİBANY
Keyword(s):  
Developable Surface ◽  
Curvature Coordinates ◽  
Line Of Curvature

An approach for designing a developable surface through a given line of curvature

2013 ◽  
Vol 45 (3) ◽  
pp. 621-627 ◽  
Author(s):  
Cai-Yun Li ◽  
Ren-Hong Wang ◽  
Chun-Gang Zhu
Keyword(s):  
Developable Surface ◽  
Line Of Curvature

Research on trajectory planning of non-developable surface for automated tape placement processing of composite

2017 ◽  
Vol 24 (6) ◽  
pp. 929-936
Author(s):  
Lin Liu ◽  
Jun Xiao ◽  
Yong Li
Keyword(s):  
Trajectory Planning ◽  
Local Stress ◽  
Developable Surface ◽  
Software Environment ◽  
Quantitative Characterization ◽  
Large Size ◽  
Variable Step ◽  
The Cost ◽  
Wing Panel

AbstractTape placement manufacturing process, as one of the automated forming technologies for composite material, not only substantially improves the productivity of composite component and reduces the cost of production significantly but also raises the reliability and stability of composite structure. Automated tape placement technology is mainly applied for manufacturing the fuselage and wing panel of airplane characterized by small curvature and large size. For these kinds of structural components with a non-developable surface, trajectory planning by “natural path” method could reduce the internal stress and improve the quality of composite products to a certain extent but not be optimized by quantitative characterization. On the basis of preliminary work, the theoretical model of “unnatural degree” (UD) is introduced in the first step, which could characterize the tensile and shear strain of the laying tape quantitatively. Secondly, by adjusting the iterative step and laying direction to diminish the UD, local stress could be softened in order to optimize the laying track. Ultimately, the simulation model of the non-developable surface is established under the Matlab software environment, and the “variable step-angle” algorithm is adopted to verify the adjustment effect of the tape-laying track.

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.

Developed Plate Expansion Using Geodesics

1984 ◽  
Vol 21 (04) ◽  
pp. 384-388
Author(s):  
John C. Clements
Keyword(s):  
Differential Equation ◽  
Numerical Quadrature ◽  
Developable Surface ◽  
Accuracy Control ◽  
Isometric Mapping ◽  
Variable Stepsize ◽  
Mapping Algorithm ◽  
Equation Solving ◽  
Expansion Procedure ◽  
The Relationship

This work is concerned with the application of a new isometric mapping algorithm to hull plate expansion procedures for ships with all or portions of the hull consisting of developable surfaces. The expansion procedure is based on the relationship between the ruling lines r⇀(s) generating the developable surface S⇀(s,t) and one additional geodesic g⇀(s) constructed within the surface as the solution of the differential equation det(g⇀'g⇀"n⇀) = 0 where n⇀ is the unit normal to S⇀ at g⇀. Precise accuracy control is achieved through the use of adaptive numerical quadrature and a variable stepsize differential equation solving routine.

Unfolding Method of Non-Developable Surface for Sheet-Metal Design

2014 ◽  
Vol 1022 ◽  
pp. 60-63
Author(s):  
Bo Sun ◽  
Xiao Long Wang
Keyword(s):  
Sheet Metal ◽  
Energy Method ◽  
Developable Surface ◽  
Metal Design ◽  
Unfolding Method

The organized vertex coordination of the triangular surfaces is obtained by the meshing method in the 3D kernel based developing environment. To optimize the flattened form, the energy method is used to find the reasonable unfolding scheme, which seeks the solution to minimize the deforming energy generated when flattening the triangular surfaces. This algorithm is not the fastest one, but the versatile one especially for the non-developable surface with a large bending degree. With the unfolding method and the 3D kernel environment, the 3D unfolding system for the arbitrary surface satisfying the requirements is constructed.

Design of a Two-Piece Brassiere Cup From its Data Points Toward its Automation

2020 ◽  
Author(s):  
Kotaro Yoshida ◽  
Hidefumi Wakamatsu ◽  
Eiji Morinaga ◽  
Takahiro Kubo
Keyword(s):  
Three Dimensional ◽  
Developable Surface ◽  
Two Dimensional ◽  
Trial And Error ◽  
Design Efficiency ◽  
Developable Surfaces ◽  
Data Points ◽  
Dimensional Shape ◽  
The Given ◽  
Three Dimensional Shape

Abstract A method to design the two-dimensional shapes of patterns of two piece brassiere cup is proposed when its target three-dimensional shape is given as a cloud of its data points. A brassiere cup consists of several patterns and their shapes are designed by repeatedly making a paper cup model and checking its three-dimensional shape. For improvement of design efficiency of brassieres, such trial and error must be reduced. As a cup model for check is made of paper not cloth, it is assumed that the surface of the model is composed of several developable surfaces. When two lines that consist in the developable surface are given, the surface can be determined. Then, the two-piece brassiere cup can be designed by minimizing the error between the surface and given data points. It was mathematically verified that the developable surface calculated by our propose method can reproduce the given data points which is developable surface.

scholarly journals Developable mechanisms on developable surfaces

2019 ◽  
Vol 4 (27) ◽  
pp. eaau5171 ◽  
Author(s):  
Todd G. Nelson ◽  
Trent K. Zimmerman ◽  
Spencer P. Magleby ◽  
Robert J. Lang ◽  
Larry L. Howell
Keyword(s):  
Mechanical Systems ◽  
Relative Orientation ◽  
Developable Surface ◽  
Curved Surfaces ◽  
Surface Type ◽  
Complex Tasks ◽  
Rigid Link ◽  
Developable Surfaces ◽  
Flat Sheet ◽  
Wide Range

The trend toward smaller mechanism footprints and volumes, while maintaining the ability to perform complex tasks, presents the opportunity for exploration of hypercompact mechanical systems integrated with curved surfaces. Developable surfaces are shapes that a flat sheet can take without tearing or stretching, and they represent a wide range of manufactured surfaces. This work introduces “developable mechanisms” as devices that emerge from or conform to developable surfaces. They are made possible by aligning hinge axes with developable surface ruling lines to enable mobility. Because rigid-link motion depends on the relative orientation of hinge axes and not link geometry, links can take the shape of the corresponding developable surface. Mechanisms are classified by their associated surface type, and these relationships are defined and demonstrated by example. Developable mechanisms show promise for meeting unfilled needs using systems not previously envisioned.

scholarly journals Designing Developable C-Bézier Surface with Shape Parameters

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 402 ◽  
Author(s):  
Caiyun Li ◽  
Chungang Zhu
Keyword(s):  
Architectural Design ◽  
Basis Functions ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Developable Surface ◽  
Rational Bézier Curves ◽  
Polynomial Representations ◽  
Rational Bézier Curve ◽  
Shape Controlling ◽  
Two Parameters

Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.

scholarly journals The one-dimensional model for d-cones revisited

2016 ◽  
Vol 9 (3) ◽  
pp. 201-215 ◽  
Author(s):  
Heiner Olbermann
Keyword(s):  
Necessary Conditions ◽  
Energy Functional ◽  
Developable Surface ◽  
Dimensional Model ◽  
Bending Energy ◽  
Main Motivation ◽  
One Dimensional ◽  
Elastic Sheet ◽  
Physics Literature ◽  
The One

AbstractA d-cone is the shape one obtains when pushing an elastic sheet at its center into a hollow cylinder. In a simple model, one can treat the elastic sheet in the deformed configuration as a developable surface with a singularity at the “tip” of the cone. In this approximation, the renormalized elastic energy is given by the bending energy density integrated over some annulus in the reference configuration. The thus defined variational problem depends on the indentation ${{h}}$ of the sheet into the cylinder. This model has been investigated before in the physics literature; the main motivation for the present paper is to give a rigorous version of some of the results achieved there via formal arguments. We derive the Gamma-limit of the energy functional as ${{h}}$ is sent to 0. Furthermore, we analyze the minimizers of the limiting functional, and list a number of necessary conditions that they have to fulfill.

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