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

Developability Approximation for Neural Implicits through Rank Minimization

Developability refers to the process of creating a surface without any tearing or shearing from a two-dimensional plane. It finds practical applications in the fabrication industry. An essential characteristic of a developable 3D surface is its zero Gaussian curvature, which means that either one or both of the principal curvatures are zero. This paper introduces a method for reconstructing an approximate developable surface from a neural implicit surface. The central idea of our method involves incorporating a regularization term that operates on the second-order derivatives of the neural implicits, effectively promoting zero Gaussian curvature. Implicit surfaces offer the advantage of smoother deformation with infinite resolution, overcoming the high polygonal constraints of state-of-the-art methods using discrete representations. We draw inspiration from the properties of surface curvature and employ rank minimization techniques derived from compressed sensing. Experimental results on both developable and non-developable surfaces, including those affected by noise, validate the generalizability of our method

Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining

We propose a new algorithm to detect patches of free-form surfaces that can be well approximated by envelopes of a rotational cone under a rigid body motion. These conical envelopes are a preferable choice from the manufacturing point of view as they are, by-definition, manufacturable by computer numerically controlled (CNC) machining using the efficient flank (peripheral) method with standard conical tools. Our geometric approach exploits multi-valued vector fields that consist of vectors in which the point-surface distance changes linearly. Integrating such vector fields gives rise to a family of integral curves, and, among them, linear segments that further serve as conical axes are quickly extracted. The lines that additionally admit tangential motion of the associated cone along the reference geometry form a set of candidate lines that are sequentially clustered and ordered to initialize motions of a rigid truncated cone. We validate our method by applying it on synthetic examples with exact envelopes, recovering correctly the exact solutions, and by testing it on several benchmark industrial datasets, detecting manufacturable conical envelope patches within fine tolerances

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