Flexibility and rigidity of frameworks consisting of triangles and parallelograms

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications Pub Date : 2023-10-05 DOI:10.1016/j.comgeo.2023.102055

Georg Grasegger , Jan Legerský

引用次数: 0

Abstract

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.

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三角形和平行四边形框架的柔韧性和刚度

框架是一个（可能是无限的）图形，其顶点在平面内实现，如果它可以在保持边长的情况下连续变形，则称为柔性框架。我们重点研究 4 循环构成平行四边形的框架的灵活性。对于本文考虑的这一类框架（允许三角形），我们证明了以下几点是等价的：柔性、无穷小柔性、存在至少两类基于 3 循环和 4 循环的等价关系以及是笛卡尔积图的非三维子图。我们研究了该问题的算法方面和旋转对称版本。结果将在由规则多边形的棋盘格得到的框架上加以说明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.