Voronoi diagrams, quasi-triangulations, and beta-complexes are powerful for solving spatial problems among spherical particles with different radii. However, a quasi-triangulation, and thus a beta-complex as well, can be a non-simplicial complex due to an anomaly condition. While a beta-complex is straightforward to use when it is a simplicial complex, it may not seem obvious if it is not. In this paper, we report the experimental statistics of showing the frequency of anomaly case occurrences in both quasi-triangulations and beta-complexes of molecular structures and randomly generated models in three-dimension. The experiment was based on 100 molecular structures from the protein data bank (PDB) and four random sets where each set consists of 100 models of three-dimensional spheres. Anomalies extremely rarely occur in molecular structures and rarely occur even in random sphere sets.
{"title":"Anomaly Occurrences in Quasi-triangulations and Beta-complexes","authors":"Donguk Kim, Youngsong Cho, Deok-Soo Kim","doi":"10.1109/ISVD.2013.18","DOIUrl":"https://doi.org/10.1109/ISVD.2013.18","url":null,"abstract":"Voronoi diagrams, quasi-triangulations, and beta-complexes are powerful for solving spatial problems among spherical particles with different radii. However, a quasi-triangulation, and thus a beta-complex as well, can be a non-simplicial complex due to an anomaly condition. While a beta-complex is straightforward to use when it is a simplicial complex, it may not seem obvious if it is not. In this paper, we report the experimental statistics of showing the frequency of anomaly case occurrences in both quasi-triangulations and beta-complexes of molecular structures and randomly generated models in three-dimension. The experiment was based on 100 molecular structures from the protein data bank (PDB) and four random sets where each set consists of 100 models of three-dimensional spheres. Anomalies extremely rarely occur in molecular structures and rarely occur even in random sphere sets.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126184800","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}
Map generalization leads to simplified maps that are needed for specific applications. However, in the map generalization process [18], the processing of the map objects and the operations applied to achieve this simplified map are usually lost. This is due to the transaction processing systems implemented in commercial GIS systems. In this research, we used the Voronoi spatial data model for map generalizations. We were able to demonstrate that the map generalization does not affect only spatial objects (points, lines or polygons), but also the events corresponding to the creation and modification of map objects, together with their temporal and spatial adjacency relationships. In this paper, we present new solutions to the problems of spatio-temporal generalizations using the hierarchical Voronoi spatio-temporal data structure. The application of the hierarchical Voronoi data structure presented in this research is in spatio-temporal map generalization, which is needed for reasoning about dynamic aspects of the world, primarily about actions, events and processes. This provides an advance in the domain of map generalization as we are able to deal not only with the cartographic objects, but also their spatio-temporal characteristics and their dynamic behaviour.
{"title":"Spatio-temporal Map Generalizations with the Hierarchical Voronoi Data Structure","authors":"D. Mioc, F. Anton, C. Gold, B. Moulin","doi":"10.1109/ISVD.2013.19","DOIUrl":"https://doi.org/10.1109/ISVD.2013.19","url":null,"abstract":"Map generalization leads to simplified maps that are needed for specific applications. However, in the map generalization process [18], the processing of the map objects and the operations applied to achieve this simplified map are usually lost. This is due to the transaction processing systems implemented in commercial GIS systems. In this research, we used the Voronoi spatial data model for map generalizations. We were able to demonstrate that the map generalization does not affect only spatial objects (points, lines or polygons), but also the events corresponding to the creation and modification of map objects, together with their temporal and spatial adjacency relationships. In this paper, we present new solutions to the problems of spatio-temporal generalizations using the hierarchical Voronoi spatio-temporal data structure. The application of the hierarchical Voronoi data structure presented in this research is in spatio-temporal map generalization, which is needed for reasoning about dynamic aspects of the world, primarily about actions, events and processes. This provides an advance in the domain of map generalization as we are able to deal not only with the cartographic objects, but also their spatio-temporal characteristics and their dynamic behaviour.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128988566","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}
Summary form only given. An appropriate concept for describing an arbitrary discrete atomic structure is the Delone set (or an (r,R)-system). Structures with long-range order such as crystals involves a concept of the space group as well. A mathematical model of an ideal monocrystalline matter is defined now as a Delone set which is invariant with respect to some space group. One should emphasize that under this definition the wellknown periodicity of crystal in all 3 dimensions is not an additional requirement. By the celebrated Schoenflies-Bieberbach theorem, any space group contains a translational subgroup with a finite index. Thus, a mathematical model of an ideal crystal uses two concepts: a Delone set (which is of local character) and a space group (which is of global character). Since the crystallization is a process which results from mutual interaction of just nearby atoms, it is believed (L. Pauling, R. Feynmann et al) that the long-range order of atomic structures of crystals (and quasi-crystals too) comes out local rules restricting the arrangement of nearby atoms. However, before 1970s there were no whatever rigorous results until Delone and his students (Dolbilin, Stogrin, Galiulin) initiated developing the local theory of crystals. The main aim of this theory was (and is) rigorous derivation of space group symmetry of a crystalline structure from the pair-wise identity of local arrangements around each atoms. To some extent, it is analogous to that as, in due time, it was rigorously proved that space group symmetry contains a translational subgroup. In the talk it is supposed to expose some results on local rules for crystals obtained by Delone, Dolbilin, Stogrin, and their followers and to outline the frontier between crystalline and quasi-crystalline local rules.
只提供摘要形式。描述任意离散原子结构的合适概念是Delone集(或(r, r)-系统)。具有长程有序的结构，如晶体，也涉及到空间群的概念。一个理想单晶物质的数学模型现在被定义为一个关于某个空间群不变的Delone集合。应该强调的是，在这个定义下，众所周知的晶体在所有三维空间的周期性并不是一个额外的要求。根据著名的Schoenflies-Bieberbach定理，任何空间群都包含一个具有有限指标的平移子群。因此，理想晶体的数学模型使用两个概念:Delone集合(具有局部特征)和空间群(具有全局特征)。由于结晶是邻近原子相互作用的结果，因此人们认为(L. Pauling, R. Feynmann等)晶体(以及准晶体)的原子结构的长程顺序产生了限制邻近原子排列的局部规则。然而，在20世纪70年代之前，没有任何严谨的结果，直到Delone和他的学生(Dolbilin, Stogrin, galulin)开始发展晶体的局部理论。该理论的主要目的是(现在也是)从每个原子周围局部排列的成对同一性中严格推导出晶体结构的空间群对称性。在某种程度上，它类似于，在适当的时候，严格证明了空间群对称包含一个平移子群。在演讲中，我们将揭示Delone, Dolbilin, Stogrin等人关于晶体局部规则的一些结果，并概述晶体和准晶体局部规则之间的界限。
{"title":"Delone Sets and Polyhedral Tilings: Local Rules and Global Order","authors":"N. Dolbilin","doi":"10.1109/ISVD.2013.22","DOIUrl":"https://doi.org/10.1109/ISVD.2013.22","url":null,"abstract":"Summary form only given. An appropriate concept for describing an arbitrary discrete atomic structure is the Delone set (or an (r,R)-system). Structures with long-range order such as crystals involves a concept of the space group as well. A mathematical model of an ideal monocrystalline matter is defined now as a Delone set which is invariant with respect to some space group. One should emphasize that under this definition the wellknown periodicity of crystal in all 3 dimensions is not an additional requirement. By the celebrated Schoenflies-Bieberbach theorem, any space group contains a translational subgroup with a finite index. Thus, a mathematical model of an ideal crystal uses two concepts: a Delone set (which is of local character) and a space group (which is of global character). Since the crystallization is a process which results from mutual interaction of just nearby atoms, it is believed (L. Pauling, R. Feynmann et al) that the long-range order of atomic structures of crystals (and quasi-crystals too) comes out local rules restricting the arrangement of nearby atoms. However, before 1970s there were no whatever rigorous results until Delone and his students (Dolbilin, Stogrin, Galiulin) initiated developing the local theory of crystals. The main aim of this theory was (and is) rigorous derivation of space group symmetry of a crystalline structure from the pair-wise identity of local arrangements around each atoms. To some extent, it is analogous to that as, in due time, it was rigorously proved that space group symmetry contains a translational subgroup. In the talk it is supposed to expose some results on local rules for crystals obtained by Delone, Dolbilin, Stogrin, and their followers and to outline the frontier between crystalline and quasi-crystalline local rules.","PeriodicalId":344701,"journal":{"name":"2013 10th International Symposium on Voronoi Diagrams in Science and Engineering","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130023959","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：