3 edition of **Continuous deformation of a developable surface** found in the catalog.

Continuous deformation of a developable surface

Zhiping Xu

- 316 Want to read
- 3 Currently reading

Published
**1996**
by University of Texas at Arlington, Dept. of Mathematics in Arlington, Tex
.

Written in English

- Surfaces, Ruled,
- Geometry, Differential,
- Isometrics (Mathematics)

**Edition Notes**

Statement | Chih-Ping Hsu. |

Series | Technical report / University of Texas at Arlington, Dept. of Mathematics -- #310., Technical report (University of Texas at Arlington. Dept. of Mathematics) -- no. 310. |

Contributions | University of Texas at Arlington. Dept. of Mathematics. |

The Physical Object | |
---|---|

Pagination | 10, [4] leaves : |

Number of Pages | 10 |

ID Numbers | |

Open Library | OL17541262M |

OCLC/WorldCa | 39155918 |

a continuous deformation: in our example the linking number of any two curves bounding a developable strip, which is invariant under isotopy, is di erent from the linking number of two concentric circles [15]. As for an outline of the content, x2 settles the notation about some natural. A developable surface is a geometric shape that can be laid out into a flat surface without stretching or tearing. The three types of developable surfaces are cylinder, cone and plane, and their corresponding projections are called cylindrical, conical and planar. Projections can be further categorized based on their point(s) of contact.

extreme deformation, external force or large timestep size. And we can readily handle various user constraints in Euclidean space. Index Terms—Developable Surface, Ribbon Simulation, Reduced Conﬁguration F 1 INTRODUCTION Developable surfaces are ubiquitous in our daily life. Although their continuous properties have been well understood [1]. Surface Book comes with great built-in apps, and you can find more apps in Microsoft Store. Processor. The 6th Gen Intel® Core i5 or i7 processor provides the speed and power you need for smooth, fast performance. Memory and storage. Choose from 8 GB of RAM with GB storage to 16 GB of RAM with GB storage.

Keywords: curved fold, developable surface, computational origami, architectural geometry, industrial design. 1 Introduction This paper is an excerpt from [Kilian et al. ]. More details on curved folding can be found in the aforementioned paper. Developable surfaces appear naturally when spatial objects are. The surface obtained by such a deformation is termed developable surface, which tra-ditionally implies a smooth, i.e., C1 continuous, developable surface. C1 developable surfaces have been used for architectural and other industrial designs since they can be produced by simply bending the surfaces of continuous sheet materials.

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The advantages of our method are twofold: by working with a continuous solution space rather than a restricted one composed of sample points, it can find mappings between input curves achieving the maximal developability; the resulting surface is directly a continuous surface and does not depend on any explicit data by: 1.

Developable surfaces have several practical applications. Developable Mechanisms are mechanisms that conform to a developable surface and can exhibit motion (deploy) off the surface. Many cartographic projections involve projecting the Continuous deformation of a developable surface book to a developable surface and then "unrolling" the surface into a region on the plane.

Since they may be constructed by bending a flat sheet, they are. Estimation of Non-rigid Surface Deformation Using Developable Surface Model Our Book Flip-ping Scanning is a new method of scanning large stacks of paper while the user performs a continuous.

We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications.

The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the by: of 3D reconstruction for book digitization and augmented reality.

Indeed, the shape of a smoothly deformed piece of paper can be very well modeled by a developable surface. Most of the existing developable surface parameterizations do not handle boundaries or are. principal curvature value of the bilayer surface is proportional to 2θ. sec With this rule, we can design more complex developable surfaces.

Deformation design of a complex surface. Let’s take a conical surface r =(ucosv,usinv,ucotγ) as an example, where u, v are two parameters of the cone, and is the half vertex angle γ of the cone. In fact, there is at most one developable surface between any two curves and there may be none.

There are, however, infinitely many warped surfaces, each warped in a different way. Figure 2 shows two edges, specifically co-axial identical circles.

On the left is the only developable surface between these two edges, a right circular cylinder. Deformation Plot. To view a deformation plot of the flattened surface, right-click the surface and click Deformation Plot.

The deformation plot shows the areas on the flattened surface with the highest levels of stretch and compression. You can mouse over the surface to. The key idea is to interleave standard physically based simulation steps with procedural generation of a piecewise continuous developable surface.

The resulting hybrid surface model captures new singular points dynamically appearing during the crumpling process, mimicking the effect of paper fiber fracture. Surface development is the process of mapping a given 3D surface into a 2D shape.

The usual objective is to achieve this mapping while preserving isometry, or, for non-developable surfaces, to. Although the two principal curvatures k 1 (p) and k 2 (p) of a surface at a point p are modified during isometric deformation, the Gaussian curvature G S (p) is constant (this is a consequence of Gauss's Theorema Egregium, which states that the Gaussian curvature is invariant under local isometry; see Do Carmo () for instance).

In the present case, this implies that the. A huge number of investigations on developable surfaces have been made, but the problem of analysis of plastic deformation during bending was usually out of the path.

It is widespread in practice to make these surfaces with the means of bending. Special attention is paid to controlling the curve of regression. Keywords: computer aided geometric design, surface approximation, developable surface, dual representation, NURBS 1 Introduction A developable surface is a surface which can be unfolded (developed) into a.

Next: Lines of curvature Up: Inflection lines of Previous: Inflection lines of Contents Index Differential geometry of developable surfaces A ruled surface is a curved surface which can be generated by the continuous motion of a straight line in space along a space curve called a directrix.

This straight line is called a generator, or ruling, of the surface. I often felt that $ K = 0$ animation that morphs between developable surfaces, viz, portions of a disk, a cone and a developable helicoid are easier to formulate for such deformation but I find no such morph animations in text books of differential geometry that I could access.

Keywords: cloth, developable surface, ﬁnite elements, constraints 1 Introduction Many deformable surfaces, ranging from most types of cloth to pa-per [Kergosien et al. ; Bo and Wang ] and stiffer materi-als, are well approximated as developable: they bend out-of-plane but do not visibly stretch or compress in-plane.

Even for materials. This paper formulates the developable surface design problem in an optimal control setting. Given a regular curve b t on the unit sphere corresponding to a one-parameter family of rulings, and two base curve endpoints a 0, a 1 ∈ R 3, we consider the problem of constructing a base curve a t such that a t 0 = a 0, a t 1 = a 1, and the resulting surface f s, t = a t + sb t is developable.

ther, a combined primal-dual surface representation enables us to robustly and quickly solve approximation problems. Additional Key Words and Phrases: interactive design, computational dif-ferential geometry, developable surface, spline surface, origami, curved folding, isometric deformation, digital reconstruction, constraint solving.

We present a new discretization for the physics-based animation of developable surfaces. Constrained to not deform at all in-plane but free to bend out-of-plane, these are an excellent approximation for many materials, including most cloth, paper, and stiffer materials.

ing, or more speciﬁcally, continuous smooth deformation of a piece of paper without creasing, while maintaining its size, i.e., allowing no size stretching or shrinking. Under smooth bending, paper, like metal sheets, can be assumed to be inextensible, and therefore assumes the shape of a developable surface, also called a developable.

While. Continuous deformation of developable surfaces and Kleing geometric analysis. We analyze the developable surfaces with the Gauss-Codazzi equation and construct a theoretical tool to investigate the transition process of the folds by examining the solutions around the singularities (folds).The key idea is to interleave standard physically-based simulation steps with procedural generation of a piecewise continuous developable surface.

The resulting hybrid surface model captures new singular points dynamically appearing during the crumpling process, mimicking the effect of paper fiber fracture.Developable sentence examplesthis algorithm is complex but very useful, it is suited to Developable surface and complex surfacecreating a rational, practicable and Developable way of life is the main point that industry design solvesthe development and utilization of content resources of physical education curriculu.