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

A Survey of Developable Surfaces: From Shape Modeling to Manufacturing

Developable surfaces are commonly observed in various applications such as architecture, product design, manufacturing, and mechanical materials, as well as in the development of tangible interaction and deformable robots, with the characteristics of easy-to-product, low-cost, transport-friendly, and deformable. Transforming shapes into developable surfaces is a complex and comprehensive task, which forms a variety of methods of segmentation, unfolding, and manufacturing for shapes with different geometry and topology, resulting in the complexity of developable surfaces. In this paper, we reviewed relevant methods and techniques for the study of developable surfaces, characterize them with our proposed pipeline, and categorize them based on digital modeling, physical modeling, interaction, and application. Through the analysis to the relevant literature, we also discussed some of the research challenges and future research opportunities.Comment: 20 pages, 24 figures, Author submitted manuscrip

Discrete Differential Geometry of Thin Materials for Computational Mechanics

Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation

State of the Art on Stylized Fabrication

© 2018 The Authors Computer Graphics Forum © 2018 The Eurographics Association and John Wiley & Sons Ltd. Digital fabrication devices are powerful tools for creating tangible reproductions of 3D digital models. Most available printing technologies aim at producing an accurate copy of a tridimensional shape. However, fabrication technologies can also be used to create a stylistic representation of a digital shape. We refer to this class of methods as ‘stylized fabrication methods’. These methods abstract geometric and physical features of a given shape to create an unconventional representation, to produce an optical illusion or to devise a particular interaction with the fabricated model. In this state-of-the-art report, we classify and overview this broad and emerging class of approaches and also propose possible directions for future research

State of the art on stylized fabrication

© 2019 Copyright held by the owner/author(s). Digital fabrication devices are powerful tools for creating tangible reproductions of 3D digital models. Most available printing technologies aim at producing an accurate copy of a tridimensional shape. However, fabrication technologies can also be used to create a stylistic representation of a digital shape. We refer to this class of methods as stylized fabrication methods. These methods abstract geometric and physical features of a given shape to create an unconventional representation, to produce an optical illusion, or to devise a particular interaction with the fabricated model. In this course, we classify and overview this broad and emerging class of approaches and also propose possible directions for future research

Designing aesthetically pleasing freeform surfaces in a computer environment

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Architecture, February 2001.Includes bibliographical references (p. 151-160).Statement: If computational tools are to be employed in the aesthetic design of freeform surfaces, these tools must better reflect the ways in which creative designers conceive of and develop such shapes. In this thesis, I studied the design of aesthetically constrained freeform surfaces in architecture and industrial design, formulated a requirements list for a computational system that would aid in the creative design of such surfaces, and implemented a subset of the tools that would comprise such a system. This work documents the clay modeling process at BMW AG., Munich. The study of that process has led to a list of tools that would make freeform surface modeling possible in a computer environment. And finally, three tools from this system specification have been developed into a proof-of-concept system. Two of these tools are sweep modification tools and the third allows a user to modify a surface by sketching a shading pattern desired for the surface. The proof-of-concept tools were necessary in order to test the validity of the tools being presented and they have been used to create a number of example objects. The underlying surface representation is a variational expression which is minimized using the finite element method over an irregular triangulated mesh.by Evan P. Smyth.Ph.D

Modeling of shell forming by line heating

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 2000.Includes bibliographical references (leaves 172-179).Metal forming by a moving heat source is an efficient and economical method for forming flat metal plates into doubly curved shapes. This thesis proposes an FEM model for three dimensional thermo-mechanical simulation of the process of shell forming by line heating. Since the heat flux is focused on a small area under the heat source, a rezoning technique is developed to reduce computation time in three-dimensional numerical simulation. This involves dynamic remeshing of the metal plate so that the area directly under the heat source is densely meshed while other areas are sparsely meshed. A simplified model is also developed which is based on semi-analytical thermal analysis and idealization of plastic zone during line heating. This simplified model is useful in real-time control of the forming process since the computation time can be greatly reduced. The two thermo-mechanical models lead to a better understanding of the line heating mechanism and more accurate and efficient prediction of the deformation of metal plates. Based on these two models, parametric studies of the edge effects, heat input, heat source velocity, spot size, heat loss coefficients, etc. are performed, and nondimensional parameters which control the bending angle are derived. Finally, an algorithm for surface development for heating path planning is developed. This algorithm minimizes the strains from the doubly curved surface to its planar development. Compared with conventional surface development methods, this algorithm takes into account the characteristics of the process of forming by line heating. This surface development algorithm lays the basis for heating path determination. Based on the developed algorithms and models, we will be able to not only determine the heating paths, but also determine the heating conditions which are necessary to form an initial flat plate into a doubly curved plate. These are critical for automation of the metal forming process.by Guoxin Yu.Ph.D

Learning and recovering 3D surface deformations

Recovering the 3D deformations of a non-rigid surface from a single viewpoint has applications in many domains such as sports, entertainment, and medical imaging. Unfortunately, without any knowledge of the possible deformations that the object of interest can undergo, it is severely under-constrained, and extremely different shapes can have very similar appearances when reprojected onto an image plane. In this thesis, we first exhibit the ambiguities of the reconstruction problem when relying on correspondences between a reference image for which we know the shape and an input image. We then propose several approaches to overcoming these ambiguities. The core idea is that some a priori knowledge about how a surface can deform must be introduced to solve them. We therefore present different ways to formulate that knowledge that range from very generic constraints to models specifically designed for a particular object or material. First, we propose generally applicable constraints formulated as motion models. Such models simply link the deformations of the surface from one image to the next in a video sequence. The obvious advantage is that they can be used independently of the physical properties of the object of interest. However, to be effective, they require the presence of texture over the whole surface, and, additionally, do not prevent error accumulation from frame to frame. To overcome these weaknesses, we propose to introduce statistical learning techniques that let us build a model from a large set of training examples, that is, in our case, known 3D deformations. The resulting model then essentially performs linear or non-linear interpolation between the training examples. Following this approach, we first propose a linear global representation that models the behavior of the whole surface. As is the case with all statistical learning techniques, the applicability of this representation is limited by the fact that acquiring training data is far from trivial. A large surface can undergo many subtle deformations, and thus a large amount of training data must be available to build an accurate model. We therefore propose an automatic way of generating such training examples in the case of inextensible surfaces. Furthermore, we show that the resulting linear global models can be incorporated into a closed-form solution to the shape recovery problem. This lets us not only track deformations from frame to frame, but also reconstruct surfaces from individual images. The major drawback of global representations is that they can only model the behavior of a specific surface, which forces us to re-train a new model for every new shape, even though it is made of a material observed before. To overcome this issue, and simultaneously reduce the amount of required training data, we propose local deformation models. Such models describe the behavior of small portions of a surface, and can be combined to form arbitrary global shapes. For this purpose, we study both linear and non-linear statistical learning methods, and show that, whereas the latter are better suited for traking deformations from frame to frame, the former can also be used for reconstruction from a single image