Computational Geometry-Theory and Applications最新文献

{"title":"Half-plane point retrieval queries with independent and dependent geometric uncertainties","authors":"Rivka Gitik, Leo Joskowicz","doi":"10.1016/j.comgeo.2023.102021","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102021","url":null,"abstract":"<div><p>This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by <em>k</em><span><span><span> real-valued parameters. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear </span>approximation of geometric uncertainty that supports parametric dependencies between </span>point locations. We present an efficient </span><span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math></span> time and space algorithm for computing the envelope of the LPGUM line that defines the half-plane query. For an exact line and an LPGUM <em>n</em> points set, we present an <span><math><mi>O</mi><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>k</mi><mo>+</mo><mi>m</mi><mi>k</mi><mo>)</mo></mrow></math></span> time query and <span><math><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow></math></span> space algorithm, where <em>m</em> is the number of LPGUM points on or above the half-plane line. For a LPGUM line and an exact points set, we present a <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mo>(</mo><mi>k</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>ε</mi></mrow></mfrac><mo>+</mo><mi>m</mi><mo>)</mo></mrow></math></span> time and <span><math><mi>O</mi><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>ε</mi></mrow></mfrac><mo>)</mo></mrow></math></span><span> space approximation algorithm, where </span><span><math><mn>0</mn><mo><</mo><mi>ε</mi><mo>≤</mo><mn>1</mn></math></span> is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mo>(</mo><mi>k</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>k</mi><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow><mrow><mi>ε</mi></mrow></mfrac><mo>+</mo><mi>m</mi><mi>k</mi><mo>)</mo></mrow></math></span> and <span><math><mi>O</mi><mrow><mo>(</mo><mi>m</mi><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mo>(</mo><mi>k</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>k</mi><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow><mrow><mi>ε</mi></mrow></mfrac><mo>)</mo></mrow></math></span> time query and <span><math><mi>O</mi><mrow><mo>(</mo><m","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"115 ","pages":"Article 102021"},"PeriodicalIF":0.6,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791616","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":"Angles of arc-polygons and Lombardi drawings of cacti","authors":"David Eppstein,&nbsp;Daniel Frishberg,&nbsp;Martha C. Osegueda","doi":"10.1016/j.comgeo.2023.101982","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101982","url":null,"abstract":"<div><p>We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are ≤<em>π</em>. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101982"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795350","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":"Bottleneck matching in the plane","authors":"Matthew J. Katz ,&nbsp;Micha Sharir","doi":"10.1016/j.comgeo.2023.101986","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101986","url":null,"abstract":"<div><p><span>We present a randomized algorithm that with high probability finds a bottleneck matching in a set of </span><span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>ℓ</mi></math></span> points in the plane. The algorithm's running time is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ω</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><mi>ω</mi><mo>&gt;</mo><mn>2</mn></math></span> is a constant such that any two <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices can be multiplied in time <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>. The state of the art in fast matrix multiplication allows us to set <span><math><mi>ω</mi><mo>=</mo><mn>2.3728596</mn></math></span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101986"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795348","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":"On the geometric priority set cover problem","authors":"Aritra Banik ,&nbsp;Rajiv Raman ,&nbsp;Saurabh Ray","doi":"10.1016/j.comgeo.2023.101984","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101984","url":null,"abstract":"<div><p><span>We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor </span>approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101984"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795351","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":"Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces","authors":"Kyungjin Cho,&nbsp;Eunjin Oh","doi":"10.1016/j.comgeo.2023.101981","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101981","url":null,"abstract":"<div><p>In this paper, we present a linear-time approximation scheme for <em>k</em>-means clustering of <em>incomplete</em> data points in <em>d</em>-dimensional Euclidean space. An <em>incomplete</em> data point with <span><math><mi>Δ</mi><mo>&gt;</mo><mn>0</mn></math></span><span><span> unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the </span>Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for </span><em>k</em>-means clustering of <em>n</em> axis-parallel affine subspaces of dimension Δ that yields an <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></math></span>-approximate solution in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>d</mi><mo>)</mo></math></span> time. The constants hidden behind <span><math><mi>O</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> depend only on <span><math><mi>Δ</mi><mo>,</mo><mi>ϵ</mi></math></span> and <em>k</em>. This improves the <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mo>)</mo></math></span>-time algorithm by Eiben et al. (2021) <span>[7]</span> by a factor of <em>n</em>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101981"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49837673","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":"Colouring bottomless rectangles and arborescences","authors":"Jean Cardinal ,&nbsp;Kolja Knauer ,&nbsp;Piotr Micek ,&nbsp;Dömötör Pálvölgyi ,&nbsp;Torsten Ueckerdt ,&nbsp;Narmada Varadarajan","doi":"10.1016/j.comgeo.2023.102020","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102020","url":null,"abstract":"<div><p>We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the <em>polychromatic k-colouring number</em> <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. This number is the smallest <em>m</em> such that any collection of bottomless rectangles can be <em>k</em>-coloured so that any <em>m</em>-fold covered point is covered by all <em>k</em> colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> is linear in <em>k</em>. We present the lower bound <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></math></span> for general families.</p><p>We also investigate <em>semi-online</em> colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers <span><math><mi>m</mi><mo>,</mo><mi>k</mi></math></span>, there is no semi-online algorithm that can <em>k</em>-colour bottomless rectangles presented in that order, so that any <em>m</em>-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"115 ","pages":"Article 102020"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791319","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":"Partial matchings induced by morphisms between persistence modules","authors":"R. Gonzalez-Diaz,&nbsp;M. Soriano-Trigueros,&nbsp;A. Torras-Casas","doi":"10.1016/j.comgeo.2023.101985","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101985","url":null,"abstract":"<div><p>We study how to obtain partial matchings using the block function <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span>, induced by a morphism <em>f</em> between persistence modules. <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span>, and provide a way of obtaining <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> using matrix operations.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101985"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795349","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":"Shortcut hulls: Vertex-restricted outer simplifications of polygons","authors":"Annika Bonerath ,&nbsp;Jan-Henrik Haunert ,&nbsp;Joseph S.B. Mitchell ,&nbsp;Benjamin Niedermann","doi":"10.1016/j.comgeo.2023.101983","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101983","url":null,"abstract":"<div><p>Let <em>P</em> be a polygon and <span><math><mi>C</mi></math></span> a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of <em>P</em>. A shortcut hull of <em>P</em> is another polygon that encloses <em>P</em> and whose oriented boundary is composed of elements from <span><math><mi>C</mi></math></span>. We require <em>P</em><span> and the output shortcut hull to be weakly simple polygons<span>, which we define as a generalization of simple polygons. Shortcut hulls find their application in cartography, where a common task is to compute simplified representations of area features. We aim at a shortcut hull that has a small area and a small perimeter. Our optimization objective is to minimize a convex combination of these two criteria. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via computation of shortest paths. For the more challenging case in which the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for the schematization of polygons.</span></span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101983"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795346","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":"Rectangle stabbing and orthogonal range reporting lower bounds in moderate dimensions","authors":"Peyman Afshani,&nbsp;Rasmus Killmann","doi":"10.1016/j.comgeo.2022.101959","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101959","url":null,"abstract":"<div><p>We study the orthogonal range reporting and rectangle stabbing problems in moderate dimensions, i.e., when the dimension is <span><math><mi>c</mi><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for some constant <em>c</em>. In orthogonal range reporting, the input is a set of <em>n</em> points in <em>d</em> dimensions, and the goal is to store these <em>n</em><span> points in a data structure such that given a query rectangle, we can report all the input points contained in the rectangle. The rectangle stabbing problem is the “dual” problem where the input is a set of rectangles, and the query is a point.</span></p><p>Our main result is the following: assume using <span><math><mi>S</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> space, we can solve either problem in <span><math><mi>d</mi><mo>=</mo><mi>c</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> dimensions, <span><math><mi>c</mi><mo>≥</mo><mn>4</mn></math></span>, using <span><math><mi>Q</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span><span> time in the pointer machine model of computation where </span><em>t</em> is the output size. Then, we show that if the query time is small, that is, <span><math><mi>Q</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>γ</mi></mrow></msup></math></span>, for <span><math><mi>γ</mi><mo>≥</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mo>+</mo><mi>log</mi><mo>⁡</mo><mi>c</mi></mrow></mfrac></math></span>, then the space must be <span><math><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>γ</mi></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><msqrt><mrow><mi>c</mi><mi>γ</mi></mrow></msqrt><mo>/</mo><mi>e</mi><mo>−</mo><mi>o</mi><mo>(</mo><msqrt><mrow><mi>c</mi><mi>γ</mi></mrow></msqrt><mo>)</mo></mrow></msup><mo>)</mo></mrow></math></span><span>. Interestingly, we obtain this lower bound using a non-constructive method, and we show the existence of some codes that generalize a specific aspect of error correction codes. Our result overcomes the shortcomings of the previous lower bounds in the pointer machine model for non-constant dimension </span><span>[3]</span>, <span>[4]</span>, <span>[5]</span>, <span>[13]</span>, as the previous results could not be extended for <span><math><mi>d</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msqrt><mo>)</mo></math></span>.</p><p>The only known lower bounds for rectangle stabbing, when the dimension is non-constant, are based on conditional lower bounds upon the best-known results on CNF-SAT <span>[21]</span>. Therefore, our lower bound is the first non-trivial unconditional lower bound for orthogonal range reporting and rectangle stabbing with non-constant dimension.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"111 ","pages":"Article 101959"},"PeriodicalIF":0.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49851373","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":"Unfoldings and nets of regular polytopes","authors":"Satyan L. Devadoss ,&nbsp;Matthew Harvey","doi":"10.1016/j.comgeo.2022.101977","DOIUrl":"https://doi.org/10.1016/j.comgeo.2022.101977","url":null,"abstract":"<div><p><span>Over a decade ago, it was shown that every edge unfolding of the Platonic solids<span> was without self-overlap, yielding a valid net. We consider this property for their higher-dimensional analogs, the regular polytopes. Three classes of regular polytopes exist for all dimensions (</span></span><em>n</em>-simplex, <em>n</em>-cube, <em>n</em>-orthoplex) and three additional regular polytopes appear only in four-dimensions (24-cell, 120-cell, 600-cell). It was recently proven that all unfoldings of the <em>n</em>-cube yield nets. We extend this to the <em>n</em><span><span>-simplex and the 4-orthoplex using the geometry of simplicial chains. Finally, we demonstrate failure of this property for any orthoplex of higher dimension, as well as for the 600-cell, providing </span>counterexamples. We conjecture failure for the two remaining open cases, the 24-cell and the 120-cell.</span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"111 ","pages":"Article 101977"},"PeriodicalIF":0.6,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49810277","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}