Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary of a domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis. Second, we present a robust implementation based on multi-precision arithmetic-a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The tests, conducted up to dimension 50, show that the HMC random walk outperforms HAR.
{"title":"Improved polytope volume calculations based on Hamiltonian Monte Carlo with boundary reflections and sweet arithmetics","authors":"F. Cazals, Augustin Chevallier, Sylvain Pion","doi":"10.20382/jocg.v13i1a3","DOIUrl":"https://doi.org/10.20382/jocg.v13i1a3","url":null,"abstract":"Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary of a domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis. Second, we present a robust implementation based on multi-precision arithmetic-a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The tests, conducted up to dimension 50, show that the HMC random walk outperforms HAR.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"10 1","pages":"52-88"},"PeriodicalIF":0.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83267759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $D$ be a set of $n$ pairwise disjoint unit balls in $R^d$ and $P$ the set of their centers. A hyperplane $H$ is an $m$-separator for $D$ if every closed halfspace bounded by $H$ contains at least $m$ points from $P$. This generalizes the notion of halving hyperplanes, which correspond to $n/2$-separators. The analogous notion for point sets is well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme, any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect a given set of pairwise disjoint unit balls by a hyperplane. Firstly, we present a simple linear-time algorithm to construct an $alpha n$-separator for balls in $R^d$, for any $0
{"title":"Halving balls by a hyperplane in deterministic linear time","authors":"M. Hoffmann, Vincent Kusters, Tillmann Miltzow","doi":"10.20382/JOCG.V11I1A23","DOIUrl":"https://doi.org/10.20382/JOCG.V11I1A23","url":null,"abstract":"Let $D$ be a set of $n$ pairwise disjoint unit balls in $R^d$ and $P$ the set of their centers. A hyperplane $H$ is an $m$-separator for $D$ if every closed halfspace bounded by $H$ contains at least $m$ points from $P$. This generalizes the notion of halving hyperplanes, which correspond to $n/2$-separators. The analogous notion for point sets is well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme, any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect a given set of pairwise disjoint unit balls by a hyperplane. Firstly, we present a simple linear-time algorithm to construct an $alpha n$-separator for balls in $R^d$, for any $0<alpha<1/2$, that intersects at most $cn^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $alpha$. The number of intersected balls is best possible up to the constant $c$. Secondly, we present a near-linear-time algorithm to construct an $(n/2-o(n))$-separator in $R^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in $mathbb{R}^2$ for $P$ that intersects $O(n^{(2/3)+epsilon})$ disks.We also point out how the above theorems can be generalized to more general classes of shapes, possibly with some overlap, and what are the limits of those generalizations.Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results simplify and derandomize an algorithm to construct a space decomposition used by Loffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk).","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"98 1","pages":"576-614"},"PeriodicalIF":0.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74248968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-31DOI: 10.5121/ijcga.2020.10301
Tariq Alrimawi, W. Haddad
{"title":"Media Students’ Capability to Interact with Augmented Reality and 3D Animations in Virtual Broadcast News Studios","authors":"Tariq Alrimawi, W. Haddad","doi":"10.5121/ijcga.2020.10301","DOIUrl":"https://doi.org/10.5121/ijcga.2020.10301","url":null,"abstract":"","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"10 1","pages":"1-15"},"PeriodicalIF":0.0,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44955466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a metric space $(P,dist)$ with $N$ points whose doubling dimension is a constant. We present a simple, randomized, and recursive algorithm that computes, in $O(N log N)$ expected time, the closest-pair distance in $P$. To generate recursive calls, we use previous results of Har-Peled and Mendel, and Abam and Har-Peled for computing a sparse annulus that separates the points in a balanced way.
{"title":"A Simple Randomized O(n log n)-Time Closest-Pair Algorithm in Doubling Metrics","authors":"A. Maheshwari, Wolfgang Mulzer, M. Smid","doi":"10.20382/JOCG.V11I1A20","DOIUrl":"https://doi.org/10.20382/JOCG.V11I1A20","url":null,"abstract":"Consider a metric space $(P,dist)$ with $N$ points whose doubling dimension is a constant. We present a simple, randomized, and recursive algorithm that computes, in $O(N log N)$ expected time, the closest-pair distance in $P$. To generate recursive calls, we use previous results of Har-Peled and Mendel, and Abam and Har-Peled for computing a sparse annulus that separates the points in a balanced way.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"362 1","pages":"507-524"},"PeriodicalIF":0.0,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76540359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-01DOI: 10.4230/LIPIcs.ESA.2020.1
A. K. Abu-Affash, S. Bhore, Paz Carmi, Joseph S. B. Mitchell
Given a set $P$ of $n$ red and blue points in the plane, a emph{planar bichromatic spanning tree} of $P$ is a spanning tree of $P$, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Moreover, we present a polynomial-time $(8sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8sqrt{2}lambda$.
{"title":"Planar Bichromatic Bottleneck Spanning Trees","authors":"A. K. Abu-Affash, S. Bhore, Paz Carmi, Joseph S. B. Mitchell","doi":"10.4230/LIPIcs.ESA.2020.1","DOIUrl":"https://doi.org/10.4230/LIPIcs.ESA.2020.1","url":null,"abstract":"Given a set $P$ of $n$ red and blue points in the plane, a emph{planar bichromatic spanning tree} of $P$ is a spanning tree of $P$, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Moreover, we present a polynomial-time $(8sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8sqrt{2}lambda$.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"42 1","pages":"109-127"},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84660772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Edge lengths of a graph are called flexible if there exist infinitely many non-congruent realizations of the graph in the plane satisfying these edge lengths. It has been shown recently that a graph has flexible edge lengths if and only if the graph has a special type of edge coloring called NAC-coloring. We address the question how to determine all possible proper flexible edge lengths from the set of all NAC-colorings of a graph. We do so using restrictions to 4-cycle subgraphs.
{"title":"On the Classification of Motions of Paradoxically Movable Graphs","authors":"Georg Grasegger, Jan Legerský, J. Schicho","doi":"10.20382/JOCG.V11I1A22","DOIUrl":"https://doi.org/10.20382/JOCG.V11I1A22","url":null,"abstract":"Edge lengths of a graph are called flexible if there exist infinitely many non-congruent realizations of the graph in the plane satisfying these edge lengths. It has been shown recently that a graph has flexible edge lengths if and only if the graph has a special type of edge coloring called NAC-coloring. We address the question how to determine all possible proper flexible edge lengths from the set of all NAC-colorings of a graph. We do so using restrictions to 4-cycle subgraphs.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"21 1","pages":"548-575"},"PeriodicalIF":0.0,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87357081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-31DOI: 10.5121/ijcga.2020.10101
Trendt Boe, Chris P. Carter
Photogrammetry has emerged as a leading approach for photorealistic digital replication and 3D scanning of real-world objects, particularly in areas of cinematic visual effects and interactive entertainment. While the technique generally relies on simple photography methods, the foundational practices for the field of human photogrammetry remain relatively undocumented.Human subjects are significantly more complex than still life, both in terms of photogrammetric capture, and in digital reproduction. Without the documentation of foundational practices for human subjects, there is a significant knowledge barrier for new creative practitioners to operate in the field, stifling innovation and adoption of the technique.Researchers and commercial practitioners currently working in this field continually distribute learnings and research outcomes. These learnings tend to centralise more on advanced practices such as capturing micro-geometry (skin pores), reflectance and skin distortion. However, the standard principles for building capture systems, considerations for human subjects, processing considerations and technology requirements remain elusive. The purpose of this research is to establish foundational practices for human photogrammetry systems. These practices encapsulate the underlying architectures of capture systems, through to necessary data processing for the 3D reconstruction of human subjects.Design-led research was used to construct a scale 21-camera system, designed for high-quality data capture of the human head. Due to its incredible level of surface complexity, the face was used to experiment with a variety of capture techniques and system arrangements, using several human subjects. The methods used were a result of the analysis of existing practitioners and research, refined through numerous iterations of system design.A distinct set of findings were synthesised to form a foundational architecture and blueprint for a scale, human photogrammetry multi-camera system. It covers the necessary knowledge and principles required to construct a production-ready photogrammetry system capable of consistent, high-quality capture that meets the needs of visual effects and interactive entertainment production.
{"title":"Human Photogrammetry: Foundational Techniques for Creative Practitioners","authors":"Trendt Boe, Chris P. Carter","doi":"10.5121/ijcga.2020.10101","DOIUrl":"https://doi.org/10.5121/ijcga.2020.10101","url":null,"abstract":"Photogrammetry has emerged as a leading approach for photorealistic digital replication and 3D scanning of real-world objects, particularly in areas of cinematic visual effects and interactive entertainment. While the technique generally relies on simple photography methods, the foundational practices for the field of human photogrammetry remain relatively undocumented.Human subjects are significantly more complex than still life, both in terms of photogrammetric capture, and in digital reproduction. Without the documentation of foundational practices for human subjects, there is a significant knowledge barrier for new creative practitioners to operate in the field, stifling innovation and adoption of the technique.Researchers and commercial practitioners currently working in this field continually distribute learnings and research outcomes. These learnings tend to centralise more on advanced practices such as capturing micro-geometry (skin pores), reflectance and skin distortion. However, the standard principles for building capture systems, considerations for human subjects, processing considerations and technology requirements remain elusive. The purpose of this research is to establish foundational practices for human photogrammetry systems. These practices encapsulate the underlying architectures of capture systems, through to necessary data processing for the 3D reconstruction of human subjects.Design-led research was used to construct a scale 21-camera system, designed for high-quality data capture of the human head. Due to its incredible level of surface complexity, the face was used to experiment with a variety of capture techniques and system arrangements, using several human subjects. The methods used were a result of the analysis of existing practitioners and research, refined through numerous iterations of system design.A distinct set of findings were synthesised to form a foundational architecture and blueprint for a scale, human photogrammetry multi-camera system. It covers the necessary knowledge and principles required to construct a production-ready photogrammetry system capable of consistent, high-quality capture that meets the needs of visual effects and interactive entertainment production.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"10 1","pages":"1-20"},"PeriodicalIF":0.0,"publicationDate":"2020-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47407106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the NP-completeness of the following problem. Given a set $S$ of $n$ slopes and an integer $kgeq 1$, is it possible to draw a complete graph on $k$ vertices in the plane using only slopes from $S$? Equivalently, does there exist a set $K$ of $k$ points in general position such that the slope of every segment between two points of $K$ is in $S$? We then present a polynomial algorithm for this question when $nleq 2k-c$, conditional on a conjecture of R.E. Jamison. For $n=k$, an algorithm in $mathcal{O}(n^4)$ was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.
{"title":"NP-completeness of slope-constrained drawing of complete graphs","authors":"Cédric Pilatte","doi":"10.20382/jocg.v11i1a14","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a14","url":null,"abstract":"We prove the NP-completeness of the following problem. Given a set $S$ of $n$ slopes and an integer $kgeq 1$, is it possible to draw a complete graph on $k$ vertices in the plane using only slopes from $S$? Equivalently, does there exist a set $K$ of $k$ points in general position such that the slope of every segment between two points of $K$ is in $S$? We then present a polynomial algorithm for this question when $nleq 2k-c$, conditional on a conjecture of R.E. Jamison. For $n=k$, an algorithm in $mathcal{O}(n^4)$ was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"17 1","pages":"371-396"},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73277289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study online routing algorithms on the $Theta$6-graph and the half-$Theta$6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the $Theta$6-graph (resp. half-$Theta$6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $Theta$6-graph and half-$Theta$6-graph defined on a Poisson point process. For the $Theta$6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-$Theta$6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89.
我们研究了$Theta$6图和半$Theta$6图(相当于Delaunay三角测量的一个变体)上的在线路由算法。给定$Theta$6图中的源顶点s和目标顶点t。half-$Theta$6-graph)，则存在一种确定性在线路由算法，该算法可以找到从s到t的路径，其长度最多为2st (resp。2.89 st)在最坏的情况下是最优的[Bose等人，siam J. on Computing, 44(6)]。我们提出了一种替代的，稍微简单的路由算法，在最坏的情况下是最优的，为此我们提供了在泊松点过程上定义的$Theta$6图和半$Theta$6图的平均路由比率的分析。对于$Theta$6图，我们的在线路由算法的期望路由比为1.161(当s和t是随机的)，最大期望路由比为1.22(对于固定的s和t，所有其他点都是随机的，最大期望路由比为2)，比最坏情况下的路由比要好得多。对于half-$Theta$6图，我们的无内存在线路由算法的期望路由比为1.43，最大期望路由比为1.58。我们的在线路由算法使用恒定数量的额外内存，其期望路由比为1.34，最大期望路由比为1.40。额外的内存只用于记住路线起点的坐标。这两种算法的期望路由比都比它们的最坏情况路由比2.89要好得多。
{"title":"Expected Complexity of Routing in $Theta_6$ and Half-$Theta_6$ Graphs","authors":"P. Bose, J. Carufel, O. Devillers","doi":"10.20382/JOCG.V11I1A9","DOIUrl":"https://doi.org/10.20382/JOCG.V11I1A9","url":null,"abstract":"We study online routing algorithms on the $Theta$6-graph and the half-$Theta$6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the $Theta$6-graph (resp. half-$Theta$6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $Theta$6-graph and half-$Theta$6-graph defined on a Poisson point process. For the $Theta$6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-$Theta$6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"48 1","pages":"212-234"},"PeriodicalIF":0.0,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77336642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
H. Akitaya, V. Dujmovic, D. Eppstein, Thomas C. Hull, Kshitij Jain, A. Lubiw
Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $mu:Eto{-1,1}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may emph{flip} a face $F$ of $G$ to create a new MV assignment $mu_F$ which equals $mu$ except for all creases $e$ bordering $F$, where we have $mu_F(e)=-mu(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $mu_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $mu_1$ and $mu_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $mu_1$ into $mu_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.
{"title":"Face flips in origami tessellations","authors":"H. Akitaya, V. Dujmovic, D. Eppstein, Thomas C. Hull, Kshitij Jain, A. Lubiw","doi":"10.20382/jocg.v11i1a15","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a15","url":null,"abstract":"Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $mu:Eto{-1,1}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may emph{flip} a face $F$ of $G$ to create a new MV assignment $mu_F$ which equals $mu$ except for all creases $e$ bordering $F$, where we have $mu_F(e)=-mu(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $mu_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $mu_1$ and $mu_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $mu_1$ into $mu_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"13 1","pages":"397-417"},"PeriodicalIF":0.0,"publicationDate":"2019-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82080253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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