Curve-pleated structures

Abstract: In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudo-geodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.

Authors/Presenter(s): Caigui Jiang, King Abdullah University of Science and Technology (KAUST), Saudi Arabia
Klara Mundilova, TU Wien, Austria
Florian Rist, King Abdullah University of Science and Technology (KAUST), Saudi Arabia
Johannes Wallner, TU Graz, Austria
Helmut Pottmann, King Abdullah University of Science and Technology (KAUST), Saudi Arabia

Modeling Curved Folding with Freeform Deformations

Abstract: We present a computational framework for interactive design and exploration of curved folded surfaces. In current practice, such surfaces are typically created manually using physical paper, and hence our objective is to lay the foundations for the digitalization of curved folded surface design. Our main contribution is a discrete binary characterization for folds between discrete developable surfaces, accompanied by an algorithm to simultaneously fold creases and smoothly bend planar sheets. We complement our algorithm with essential building blocks for curved folding deformations: objectives to control dihedral angles and mountain-valley assignments. We apply our machinery to build the first interactive freeform editing tool capable of modeling bending and folding of complicated crease patterns.

Authors/Presenter(s): Michael Rabinovich, ETH Zurich, Switzerland
Tim Hoffmann, Technische Universität München (TUM), Germany
Olga Sorkine-Hornung, ETH Zurich, Switzerland

Checkerboard Patterns with Black Rectangles

Abstract: Checkerboard patterns with black rectangles can be derived from quad meshes with orthogonal diagonals. First, we present an initial theoretical analysis of these quad meshes. The analysis reveals many possible applications in geometry processing and also motivates the numerical optimization for aesthetic and functional checkerboard pattern design. Second, we describe an optimization algorithm that transforms initial 2D and 3D quad meshes into quad meshes with orthogonal diagonals. Third, we present a 2D checkerboard pattern design framework based on integer programming inspired by the logo design of the 2020 Olympic games. Our results show a variety of 2D and 3D checkerboard patterns that can be derived from 2D or 3D quad meshes with orthogonal diagonals.

Authors/Presenter(s): Chi-Han Peng, King Abdullah University of Science and Technology (KAUST), Saudi Arabia
Caigui Jiang, King Abdullah University of Science and Technology (KAUST), Saudi Arabia
Peter Wonka, King Abdullah University of Science and Technology (KAUST), Saudi Arabia
Helmut Pottmann, King Abdullah University of Science and Technology (KAUST), Saudi Arabia

Chebyshev Nets from Commuting PolyVector Fields

Abstract: We propose a method for computing global Chebyshev nets on triangular meshes. We formulate the corresponding global parameterization problem in terms of commuting PolyVector fields, and design an efficient optimization method to solve it. We compute, for the first time, Chebyshev nets with automatically-placed singularities, and demonstrate the realizability of our approach using real material.

Authors/Presenter(s): Andrew O. Sageman-Furnas, Technical University of Berlin, Germany
Albert Chern, Technical University of Berlin, Germany
Mirela Ben-Chen, Technion-Israel Institute of Technology, Israel
Amir Vaxman, Utrecht University, Netherlands

Discrete Geodesic Parallel Coordinates

Abstract: Geodesic parallel coordinates are orthogonal nets on surfaces where one of the two families of parameter lines are geodesic curves. We describe a discrete version of these special surface parameterizations and show that they are very useful for specific applications, most of which are related to the design and fabrication of surfaces in architecture. With the new discrete surface models, it is easy to control strip widths between neighboring geodesics. This facilitates tasks such as cladding a surface with strips of originally straight flat material or designing geodesic gridshells or timber rib shells. It is also possible to model nearly developable surfaces. These are characterized by geodesic strips with almost constant strip widths and are used for generating shapes that can be manufactured from materials which allow for some stretching or shrinking like felt, leather, or thin wooden boards. Most importantly, we show how to constrain the strip width parameters to model a class of intrinsically symmetric surfaces. These surfaces are isometric to surfaces of revolution and can be covered with double curved panels that are produced with only a few molds when working with flexible materials like metal sheets.

Authors/Presenter(s): Hui Wang, Dalian University of Technology, TU Wien, China
Davide Pellis, TU Wien, Austria
Florian Rist, King Abdullah University of Science and Technology (KAUST), TU Wien, Austria
Helmut Pottmann, King Abdullah University of Science and Technology (KAUST), TU Wien, Austria
Christian Mueller, TU Wien, Austria