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Register a new userThe designs and deformations of rigidly and flat-foldable quadrilateral mesh origami
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Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of piecewise affine deformations that do not stretch or bend the mesh-panels. In this work, we explicitly characterize the designs and deformations of all possible rigidly and flat-foldable quadrilateral mesh origami. Our key idea is a rigidity theorem (Theorem 3.1) that characterizes compatible crease patterns surrounding a single panel and enables us to march from panel to panel to compute the pattern and its corresponding deformations explicitly. The marching procedure is computationally efficient. So we use it to formulate the inverse problem: to design a crease pattern to achieve a targeted shape along the path of its rigidly and flat-foldable motion. The initial results on the inverse problem are promising and suggest a broadly useful engineering design strategy for shape-morphing with origami.
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Origami, where two-dimensional sheets are folded into complex structures, is proving to be rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider emph{flat origami}, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases made by such foldings, called a emph{MV assignment} of the crease pattern. We establish a method to, given a flat-foldable crease pattern $C$ under certain conditions, create a planar graph $C^*$ whose 3-colorings are in one-to-one correspondence with the locally-valid MV assignments of $C$. This reduces the general, unsolved problem of enumerating locally-valid MV assignments to the enumeration of 3-colorings of graphs.
Using a mathematical model for self-foldability of rigid origami, we determine which monohedral quadrilateral tilings of the plane are uniquely self-foldable. In particular, the Miura-ori and Chicken Wire patterns are not self-foldable under our definition, but such tilings that are rotationally-symmetric about the midpoints of the tile are uniquely self-foldable.
Origami, the ancient art of folding thin sheets, has attracted increasing attention for its practical value in diverse fields: architectural design, therapeutics, deployable space structures, medical stent design, antenna design and robotics. In this survey article we highlight its suggestive value for the design of materials. At continuum level the rules for constructing origami have direct analogs in the analysis of the microstructure of materials. At atomistic level the structure of crystals, nanostructures, viruses and quasicrystals all link to simplified methods of constructing origami. Underlying these linkages are basic physical scaling laws, the role of isometries, and the simplifying role of group theory. Non-discrete isometry groups suggest an unexpected framework for the possible design of novel materials.
This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic. Each quad-mesh induces a conformal structure of the surface, and a meromorphic differential, where the configuration of singular vertices correspond to the configurations the poles and zeros (divisor) of the meroromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel-Jacobi condition. Inversely, if a satisfies the Abel-Jacobi condition, then there exists a meromorphic quartic differential whose equals to the given one. Furthermore, if the meromorphic quadric differential is with finite, then it also induces a a quad-mesh, the poles and zeros of the meromorphic differential to the singular vertices of the quad-mesh. Besides the theoretic proofs, the computational algorithm for verification of Abel-Jacobi condition is explained in details. Furthermore, constructive algorithm of meromorphic quartic differential on zero surfaces is proposed, which is based on the global algebraic representation of meromorphic. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a direction for quad-mesh generation using algebraic geometric approach.
Mesh autoencoders are commonly used for dimensionality reduction, sampling and mesh modeling. We propose a general-purpose DEep MEsh Autoencoder (DEMEA) which adds a novel embedded deformation layer to a graph-convolutional mesh autoencoder. The embedded deformation layer (EDL) is a differentiable deformable geometric proxy which explicitly models point displacements of non-rigid deformations in a lower dimensional space and serves as a local rigidity regularizer. DEMEA decouples the parameterization of the deformation from the final mesh resolution since the deformation is defined over a lower dimensional embedded deformation graph. We perform a large-scale study on four different datasets of deformable objects. Reasoning about the local rigidity of meshes using EDL allows us to achieve higher-quality results for highly deformable objects, compared to directly regressing vertex positions. We demonstrate multiple applications of DEMEA, including non-rigid 3D reconstruction from depth and shading cues, non-rigid surface tracking, as well as the transfer of deformations over different meshes.
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