Computational Geometry-Theory and Applications最新文献

In this study, we lay the groundwork for a systematic investigation of the rigidity and flexibility of rigid origami by using the mathematical model referred to as the panel-point model. Rigid origami is commonly known as a type of panel-hinge structure where rigid polygonal panels are connected by rotational hinges, and its motion and stability are often investigated from the perspective of its consistency constraints representing the rigidity and connection conditions of panels. In the proposed methodology, vertex coordinates are directly treated as the variables to represent the rigid origami in the panel-point model, and these variables are constrained by the conditions for the out-of-plane and in-plane rigidity of panels. This model offers several advantages including: 1) the simplicity of polynomial consistency constraints; 2) the ease of incorporating displacement boundary conditions; and 3) the straightforwardness of numerical simulation and visualization. It is anticipated that the presented theories in this article are valuable to a broad audience, including mathematicians, engineers, and architects.

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.

Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.

A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal flexibility, the existence of at least two classes of an equivalence relation based on 3- and 4-cycles and being a non-trivial subgraph of the Cartesian product of graphs. We study the algorithmic aspects and the rotationally symmetric version of the problem. The results are illustrated on frameworks obtained from tessellations by regular polygons.

Given a set of n colored points with k colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axis-parallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus A of a particular shape with maximum possible width such that A does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in $O\left(n^3\right)$ time using $O\left(n\right)$ space, in $O(k^2{n}^2\log n)$ time using $O(n\log n)$ space and in $O\left(n^3\right)$ time using $O\left(n^2\right)$ space respectively.

We define a $C\left(k\right)$ to be a family of k sets $F_1,\dots ,F_k$ such that $\text{conv}(F_i\cup F_{i+1})\cap \text{conv}(F_j\cup F_{j+1})=\varnothing$ when $\{i,i+1\}\cap \{j,j+1\}=\varnothing$ (indices are taken modulo k). We show that if $F$ is a family of compact, convex sets that does not contain a $C\left(k\right)$, then there are $k-2$ lines that pierce $F$. Additionally, we give an example of a family of compact, convex sets that contains no $C\left(k\right)$ and cannot be pierced by $\lceil \frac{k}{2}\rceil -1$ lines.

Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.

A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools from toric geometry. We introduce a link between self-stresses and Chow rings on toric varieties. More precisely, for a given rational tensegrity framework $F$, we construct a glued toric surface $X_F$. We show that the abelian group of tensegrities on $F$ is isomorphic to a subgroup of the Chow group $A^1(X_F;Q)$. In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.

We examine rectangle packing problems where only the areas $a_1,\dots ,a_n$ of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval $[\frac{1}{\gamma },\gamma ]$. In particular, we ask for the smallest possible size of a rectangle R such that, under these constraints, any collection $a_1,\dots ,a_n$ of rectangle areas of total size 1 can be packed into R. As for standard square packing problems, which are contained as a special case for $\gamma =1$, this question leads us to three different answers, depending on whether the aspect ratio of R is given or whether we may choose it either with or without knowing the areas $a_1,\dots ,a_n$. Generalizing known results for square packing problems, we provide upper and lower bounds for the size of R with respect to all three variants of the problem, which are tight at least for larger values of γ. Moreover, we show how to improve these bounds on the size of R if we restrict ourselves to instances where the largest element in $a_1,\dots ,a_n$ is bounded.

We initiate the study of a generalization of the class cover problem [Cannon and Cowen [1], Bereg et al. [2]] the generalized class cover problem, where we are allowed to misclassify some points provided we pay an associated positive penalty for every misclassified point. Two versions: single coverage and multiple coverage, of the generalized class cover problem are investigated. We study five different variants of both versions of the generalized class cover problem with axis-parallel strips and axis-parallel half-strips extending to different directions in the plane, thus extending similar work by Bereg et al. (2012) [2] on the class cover problem. We prove that the multiple coverage version of the generalize class cover problem with axis-parallel strips are in $P$, whereas the single coverage version is $\mathrm{NP}$-hard. A factor 2 approximation algorithm is provided for the later problem. The $\mathrm{APX}$-hardness result is also shown for the single coverage version. For half-strips extending to exactly one direction, both the single and multiple coverage versions can be solved in polynomial time using dynamic programming. In the case of half-strips extending to two orthogonal directions, we prove the class cover problem is $\mathrm{NP}$-hard followed by $\mathrm{APX}$-hard. This gives improve hardness results compare to Bereg et al. (2012) [2], where they proved the class cover problem with half-strips oriented in four different directions is $\mathrm{NP}$-hard. These $\mathrm{NP}$- and $\mathrm{APX}$-hardness results can directly apply to both single and multiple versions. Finally, constant factor approximation algorithms are provided for half-strips extending to more than one direction.