Computational Geometry-Theory and Applications最新文献

{"title":"On the orthogonal Grünbaum partition problem in dimension three","authors":"","doi":"10.1016/j.comgeo.2024.102149","DOIUrl":"10.1016/j.comgeo.2024.102149","url":null,"abstract":"<div><div>Grünbaum's equipartition problem asked if for any measure <em>μ</em> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> there are always <em>d</em> hyperplanes which divide <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> into <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math></span> <em>μ</em>-equal parts. This problem is known to have a positive answer for <span><math><mi>d</mi><mo>≤</mo><mn>3</mn></math></span> and a negative one for <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></span>. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for <span><math><mi>d</mi><mo>≤</mo><mn>2</mn></math></span> and there is reason to expect it to have a negative answer for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of 8<em>n</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> can be split evenly by 3 mutually orthogonal planes. To our surprise, it seems the probability that a random set of 8 points chosen uniformly and independently in the unit cube does not admit such a partition is less than 0.001.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization","authors":"","doi":"10.1016/j.comgeo.2024.102147","DOIUrl":"10.1016/j.comgeo.2024.102147","url":null,"abstract":"<div><div>Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence <em>retraction maps</em> are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and prove convergence results.</div><div>We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Largest unit rectangles inscribed in a convex polygon","authors":"","doi":"10.1016/j.comgeo.2024.102135","DOIUrl":"10.1016/j.comgeo.2024.102135","url":null,"abstract":"<div><p>We consider an optimization problem of inscribing a unit rectangle in a convex polygon. An axis-aligned unit rectangle is an axis-aligned rectangle whose horizontal sides are of length 1. A unit rectangle of orientation <em>θ</em> is a copy of an axis-aligned unit rectangle rotated by <em>θ</em> in counterclockwise direction. The goal is to find a largest unit rectangle inscribed in a convex polygon over all orientations in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>)</mo></math></span>. This optimization problem belongs to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000579/pdfft?md5=421deb79b0fe58ffb995ba93bffa3330&pid=1-s2.0-S0925772124000579-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142020759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Packing unequal disks in the Euclidean plane","authors":"","doi":"10.1016/j.comgeo.2024.102134","DOIUrl":"10.1016/j.comgeo.2024.102134","url":null,"abstract":"<div><p>A packing of disks in the plane is a set of disks with disjoint interiors. This paper is a survey of some open questions about such packings. It is organized into five themes: compacity, conjugacy, density, uniformity and computability.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000567/pdfft?md5=fb180e9154b1ec63995a4bc9108b1b08&pid=1-s2.0-S0925772124000567-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved approximation for two-dimensional vector multiple knapsack","authors":"","doi":"10.1016/j.comgeo.2024.102124","DOIUrl":"10.1016/j.comgeo.2024.102124","url":null,"abstract":"<div><p>We study the <span>uniform</span> 2<span>-dimensional vector multiple knapsack</span> (2VMK) problem, a natural variant of <span>multiple knapsack</span> arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional <em>weight</em> vector and a positive <em>profit</em>, along with <em>m</em> 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.</p><p>Our main result is a <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm for 2VMK, for every fixed <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span>, thus improving the best known ratio of <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span> which follows as a special case from a result of Fleischer et al. (2011) <span><span>[6]</span></span>.</p><p>Our algorithm relies on an adaptation of the Round&amp;Approx framework of Bansal et al. (2010) <span><span>[15]</span></span>, originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to <span><math><mo>≈</mo><mi>m</mi><mo>⋅</mo><mi>ln</mi><mo>⁡</mo><mn>2</mn><mo>≈</mo><mn>0.693</mn><mo>⋅</mo><mi>m</mi></math></span> of the bins, followed by a reduction to the (1-dimensional) <span>Multiple Knapsack</span> problem for assigning items to the remaining bins.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000464/pdfft?md5=aabf82f5f8cf463934bfaf0d08024ae5&pid=1-s2.0-S0925772124000464-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the line-separable unit-disk coverage and related problems","authors":"","doi":"10.1016/j.comgeo.2024.102122","DOIUrl":"10.1016/j.comgeo.2024.102122","url":null,"abstract":"<div><p>Given a set <em>P</em> of <em>n</em> points and a set <em>S</em> of <em>m</em> disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of <em>P</em>. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of <em>P</em> by a line <em>ℓ</em>. We present an <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mo>)</mo></math></span> time algorithm for the problem. This improves the previously best result of <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>m</mi><mo>+</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of <em>S</em> are located on a line <em>ℓ</em> while points of <em>P</em> can be anywhere in the plane. Our algorithm runs in <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time, which improves the previously best result of <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>m</mi><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time. In addition, our results lead to an algorithm of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time for a half-plane coverage problem (given <em>n</em> half-planes and <em>n</em> points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. Further, if all half-planes are lower ones, our algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time while the previously best algorithm takes <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on the k-colored crossing ratio of dense geometric graphs","authors":"","doi":"10.1016/j.comgeo.2024.102123","DOIUrl":"10.1016/j.comgeo.2024.102123","url":null,"abstract":"<div><p>A <em>geometric graph</em> is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a constant <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> such that the following holds. The edges of every dense geometric graph, with sufficiently many vertices, can be colored with <em>k</em> colors, such that the number of pairs of edges of the same color that cross is at most <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mi>k</mi><mo>−</mo><mi>c</mi><mo>)</mo></math></span> times the total number of pairs of edges that cross. The case when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> and <em>G</em> is a complete geometric graph, was proved by Aichholzer et al. (2019) <span><span>[2]</span></span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000452/pdfft?md5=232d9c5eb8dccf79fd64157d664cfa52&pid=1-s2.0-S0925772124000452-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Routing on heavy path WSPD spanners","authors":"","doi":"10.1016/j.comgeo.2024.102121","DOIUrl":"10.1016/j.comgeo.2024.102121","url":null,"abstract":"<div><p>In this article, we present a construction of a spanner on a set of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> that we call a heavy path WSPD spanner. The construction is parameterized by a constant <span><math><mi>s</mi><mo>&gt;</mo><mn>2</mn></math></span> called the separation ratio. The size of the graph is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup><mi>n</mi><mo>)</mo></math></span> and the spanning ratio is at most <span><math><mn>1</mn><mo>+</mo><mn>2</mn><mo>/</mo><mi>s</mi><mo>+</mo><mn>2</mn><mo>/</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. We also show that this graph has a hop spanning ratio of at most <span><math><mn>2</mn><mi>lg</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>.</p><p>We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex <em>v</em> of the graph to store <span><math><mi>O</mi><mo>(</mo><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> bits of information, where <span><math><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is the degree of <em>v</em>. The routing ratio is at most <span><math><mn>1</mn><mo>+</mo><mn>4</mn><mo>/</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo>/</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and at least <span><math><mn>1</mn><mo>+</mo><mn>4</mn><mo>/</mo><mi>s</mi></math></span> in the worst case. The number of edges on the routing path is bounded by <span><math><mn>2</mn><mi>lg</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>.</p><p>We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension <em>λ</em>, the heavy path WSPD spanner has size <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msup><mi>n</mi><mo>)</mo></math></span> where <em>s</em> is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case.</p><p>Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most <span><math><mn>1</mn><mo>+</mo><mo>(</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mi>τ</mi></mrow><mrow><mi>τ</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>)</mo><mo>/</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo>/</mo><mo>(</mo><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> in the worst case, where <span><math><mi>τ</mi><mo>≥</mo><mn>11</mn></math></span> is a constant related to the doubling dimension.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000439/pdfft?md5=bf39cad158ed560ddaba5bed399d108b&pid=1-s2.0-S0925772124000439-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141623649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Online class cover problem","authors":"Minati De ,&nbsp;Anil Maheshwari ,&nbsp;Ratnadip Mandal","doi":"10.1016/j.comgeo.2024.102120","DOIUrl":"https://doi.org/10.1016/j.comgeo.2024.102120","url":null,"abstract":"<div><p>In this paper, we study the online class cover problem where a (finite or infinite) family <span><math><mi>F</mi></math></span> of geometric objects and a set <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> of red points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> are given a prior, and blue points from <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from <span><math><mi>F</mi></math></span> that do not cover any points of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>. The objective of the problem is to place a minimum number of objects. When <span><math><mi>F</mi></math></span> consists of axis-parallel unit squares in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, we prove that the competitive ratio of any deterministic online algorithm is <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>|</mo><mo>)</mo></math></span>, and also propose an <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>|</mo><mo>)</mo></math></span>-competitive deterministic algorithm for the problem.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141539622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Isometric deformations of discrete and smooth T-surfaces","authors":"Ivan Izmestiev,&nbsp;Arvin Rasoulzadeh,&nbsp;Jonas Tervooren","doi":"10.1016/j.comgeo.2024.102104","DOIUrl":"https://doi.org/10.1016/j.comgeo.2024.102104","url":null,"abstract":"<div><p>Quad-surfaces are polyhedral surfaces with quadrilateral faces and the combinatorics of a square grid. Isometric deformation of the quad-surfaces can be thought of as transformations that keep all the involved quadrilaterals rigid. Among quad-surfaces, those capable of non-trivial isometric deformations are identified as flexible, marking flexibility as a core topic in discrete differential geometry. The study of quad-surfaces and their flexibility is not only theoretically intriguing but also finds practical applications in fields like membrane theory, origami, architecture and robotics.</p><p>A generic quad-surface is rigid, however, certain subclasses exhibit a 1-parameter family of flexibility. One of such subclasses is the T-hedra which are originally introduced by Graf and Sauer in 1931.</p><p>This article provides a synthetic and an analytic description of T-hedra and their smooth counterparts namely, the T-surfaces. In the next step the parametrization of their isometric deformation is obtained and their deformability range is discussed. The given parametrizations and isometric deformations are provided for general T-hedra and T-surfaces. However, specific subclasses are extensively examined and explored, particularly those that encompass notable and well-known structures, including the Miura fold, surfaces of revolution and molding surfaces.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000269/pdfft?md5=f783a67dd89d4ba5ecb5b55e3219f48b&pid=1-s2.0-S0925772124000269-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140950658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}