Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Developable B-spline surface generation from control rulings

An intuitive design method is proposed for generating developable ruled B-spline surfaces from a sequence of straight line segments indicating the surface shape. The first and last line segments are enforced to be the head and tail ruling lines of the resulting surface while the interior lines are required to approximate rulings on the resulting surface as much as possible. This manner of developable surface design is conceptually similar to the popular way of the freeform curve and surface design in the CAD community, observing that a developable ruled surface is a single parameter family of straight lines. This new design mode of the developable surface also provides more flexibility than the widely employed way of developable surface design from two boundary curves of the surface. The problem is treated by numerical optimization methods with which a particular level of distance error is allowed. We thus provide an effective tool for creating surfaces with a high degree of developability when the input control rulings do not lie in exact developable surfaces. We consider this ability as the superiority over analytical methods in that it can deal with arbitrary design inputs and find practically useful results.Comment: 13 pages, 12 figrue

Non-singular developable triangular Bézier patches

We show a characterisation of developable surfaces in the form of B´ezier triangular patches. • Constructions used for rectangular patches are not useful, since they produce degenerate triangular patches. • Explicit constructions of non-degenerate developable triangular patches are provided. • Interpolation of a developable triangle between a curve c(u), the last ruling and initial velocity of the other bounding curve d(u)

Triangular Bézier Developable Patches

Developable surfaces are defined as zero gaussian curvature surfaces (intrinsically flat). That is, plane patches that are curved by just folding, rolling or cutting, but without stretching or combing. Useful for depicting steel plates in naval industry, cloth in textile industry. . . But they are difficult to include in the NURBS formulation for the zero curvature requirement

曲線折り動作のモデル化と可視化に関する研究

筑波大学 (University of Tsukuba)201

Computer-Aided Development of Shell Plates

Ship hulls and other curved shells, like gas tanks, aircraft bodies, and even clothes and shoes, offer a common difficulty in their manufacturing: it is necessary to produce them from a set of formerly plane elements. These plane elements, the raw materials like plates and fabric pieces, must be curved and assembled together to form the final product. The reverse of the forming process of these curved elements, is the map of the curved surface onto the plane, which is improperly known as development. To develop a surface, in a proper sense, is to unfold it onto the plane without stretching or bulging. This is not possible with all kinds of shapes, such as spherical and saddle surfaces. Some common developable surfaces are the conical and cylindrical ones. To form a non-developable shell requires much more work than to form an equivalent shell of developable shape. This increases the costs, the processing times and the defect content. Nevertheless, the fluid dynamists and the other designers are not always free to use developable shapes in their concepts; therefore, a pragmatic approach to the construction of curved shells has to cope with non-developable surfaces. These subjects are chiefly of an advanced mathematic nature, and the required background is too widely spread in the bibliography. Therefore the necessary mathematical results are compiled and presented in Chapter 2 - The Mathematics of Developable Surfaces, providing for a unified view of the concepts, the symbols and the nomenclature. Since the advent of the digital computer, the increasing availability of computing power enabled new methods for surface development and for developable surface definition. By examining and comparing the methods reported in the literature, CHAPTER 3 - Plate Development and Developable Surfaces provides a broad view of the surface development issues, along with the developability conditions and the technologies for the definition of developable surfaces. Given the absence of developability conditions in some areas of the shell, a number of methodologies are reported which produce a plate map onto the plane. In Chapter 4 - Concept and Implementation of an Algorithm, the concept and the implementation of a new development algorithm is described, analysed and applied to example cases. By geodesicaly mapping the surface onto the plane, this method avoids the implementation difficulties of both non- developable surfaces, and developable surfaces with ruling lines aligned in any direction. Therefore, the slightly non-developable plates, commonly found in actual ship hulls, are easily accommodated by this process, working as a map onto the plane. In Chapter 5 - Industrial Application of the Improper Geodesic Map, the user interface of the method is presented. The method provides information about the surface developability and fairness, which assists the user in the decision to develop or otherwise to take corrective measures, like re-fairing or editing of seams and butts. Results obtained from analytical plates, and comparisons with results from both a 1/10-scale electrostatic development jig, and a commercial software package, validate the method. Other results, obtained from actual ship plates, are also presented, further confirming the good accuracy of the method's developments and its good behaviour when processing non-developable plates. This method is in current use, as part of a shipyard system