V. On certain developable surfaces

1863 ◽  
Vol 12 ◽  
pp. 279-280 ◽  
Keyword(s):  
Developable Surface ◽  
Developable Surfaces

If U = 0 be the equation of a developable surface, or say a developable, then the hessian HU vanishes, not identically, but only by virtue of the equation U = 0 of the surface; that is, HU contains U as a factor, or we may write HU = U. PU. The function PU, which for the developable replaces, as it were, the hessian HU, is termed the prohessian; and since, if r be the order of U, the order of HU is 4 r —8, we have 3 r —8 for the order of the prohessian. If r =4, the order of the prohessian is also 4; and in fact, as is known, the prohessian is in this case = U.

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 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 Parameterisation of developable surfaces by asymptotic lines

1996 ◽  
Vol 54 (3) ◽  
pp. 411-421 ◽  
Author(s):  
Vitaly Ushakov
Keyword(s):  
Explicit Form ◽  
Gaussian Curvature ◽  
Developable Surface ◽  
Developable Surfaces ◽  
Asymptotic Lines

An example of a “non-developable” surface of vanishing Gaussian curvature from W. Klingenberg's textbook is considered and its place in the theory of 2-dimensional developable surfaces is pointed out. The surface is found in explicit form. Other examples of smooth developable surfaces not allowing smooth asymptotic parametrisation are analysed. In particular, Hartman and Nirenberg's example (1959) is incorrect.

scholarly journals Polar and Equatorial Aspects of Map Projections?

2019 ◽  
Vol 2 ◽  
pp. 1-6
Author(s):  
Miljenko Lapaine ◽  
Nedjeljko Frančula
Keyword(s):  
Perspective Projection ◽  
Developable Surface ◽  
Map Projection ◽  
Map Projections ◽  
Developable Surfaces

<p><strong>Abstract.</strong> There is no standard or generally accepted terminology of aspect in the theory of map projections. The term is probably derived from the concept that a graticule is produced by perspective projection of the meridians and parallels on a sphere onto a developable surface. Developable surfaces are widely accepted, and it is almost impossible to find a publication that deals with map projections in general and without developable surfaces story. If found, it usually classifies projections as cylindrical, conical and azimuthal, and applies developable surfaces to define the projection aspect. This paper explains why applying developable surfaces in the interpretation of map projections is not recommended, nor defining the aspect of all projections by the position of a midpoint as polar, equatorial, or oblique. In fact, defining a projection aspect this way is invalid in general, and obscures the fact that the aspect depends on the class to which a particular map projection belongs.</p>

Newer Theory and More Robust Algorithms for Computer-Aided Design of Developable Surfaces

2005 ◽  
Vol 42 (03) ◽  
pp. 71-79
Author(s):  
B. Konesky
Keyword(s):  
Computer Aided Design ◽  
Developable Surface ◽  
Robust Algorithms ◽  
Developable Surfaces ◽  
Space Curves ◽  
Surface Areas ◽  
C Programming ◽  
Computer Aided ◽  
Reliable Solution ◽  
Aided Design

The use of developable surfaces in design is of engineering importance because of the relative ease with which they can be manufactured. The problem of how to make surfaces developable is not new. The usual technique is by using two space curves, defining the edges of the surface. These are first created, and then a set of rulings are constructed between the space curves under the constraint of being developable. A problem with existing algorithms for designing developable surfaces is the tendency to include nondevelopable portions of the surface: areas of regression. A more reliable solution to the problem of creating a developable surface is presented. The key to the method is to define the developable surface in terms of a normal directrix. The shape of the normal directrix defines the resulting developable surface. Algorithms are defined to compute the shape of a normal directrix from a pair of space curves. Intersecting adjacent developable surfaces and generating the flat plate layouts were also accomplished. This paper presents research and development that started around 1987. The algorithms were implemented using ANSI C++ programming language and commercial computer-aided design and manufacturing (CAD and CAM) software programs.

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.

scholarly journals Flat Approximations of Surfaces Along Curves

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Shyuichi Izumiya ◽  
Saki Otani
Keyword(s):  
Orthogonal Projection ◽  
Central Projection ◽  
Developable Surface ◽  
Developable Surfaces

AbstractWe consider a developable surface tangent to a surface along a curve on the surface. We call it an osculating developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities. As a by-product, we show that a curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants constantly equal to zero.

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.

scholarly journals Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces associated with a space curve

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200617
Author(s):  
Brian Seguin ◽  
Yi-chao Chen ◽  
Eliot Fried
Keyword(s):  
Geodesic Curvature ◽  
Ruled Surface ◽  
Space Curve ◽  
Developable Surface ◽  
Absolute Value ◽  
Developable Surfaces ◽  
Alternative Construction ◽  
Tangent Developable ◽  
The Absolute

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies | k | ≤  κ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g  =  k .

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