Related papers: Computational Geometry Column 36
In this paper, we provide assembly instructions for an easy to build experimental setup in order to gain practical experience with tomography. In view of this, this paper can be seen as a complementary work to excellent mathematical…
An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main…
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 optimal calculation order of a computational graph can be represented by a set of algebraic expressions. Computational graph and algebraic expression both have close relations and significant differences, this paper looks into these…
We present a formalization of geometric instruments that considers separately geometric and arithmetic aspects of them. We introduce the concept of tool, which formalizes a physical instrument as a set of axioms representing its geometric…
We develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and results from cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First,…
We draw two incomplete, biased maps of challenges in computational complexity lower bounds.
We survey several results known on sampling in computational geometry.
In the making of origami, one starts with a piece of paper, and through a series of folds along seed points one constructs complicated three-dimensional shapes. Mathematically, one can think of the complex numbers as representing the piece…
This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is…
We suggest a diagrammatic model of computation based on an axiom of distributivity. A diagram of a decorated coloured tangle, similar to those that appear in low dimensional topology, plays the role of a circuit diagram. Equivalent diagrams…
Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal…
The concept of pointed pseudo-triangulations is defined and a few of its applications described.
We derive new algebraic equations for the folding angle relationships in completely general degree-four rigid-foldable origami vertices, including both Euclidean (developable) and non-Euclidean cases. These equations in turn lead to novel,…
This paper is a direct continuation of the paper arXiv:2401.00053. By this reason neither introductory part of the paper nor the list of references are not duplicated. However for the reader convenience, the formulas from the first paper…
"Flat origami" refers to the folding of flat, zero-curvature paper such that the finished object lies in a plane. Mathematically, flat origami consists of a continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ along…
Kirigami, art of paper cutting, enables two-dimensional sheets transforming into unique shapes which are also hard to reshape once with prescribed cutting patterns. Rare kirigami designs manipulate cuts on three-dimensional objects to…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.