Pub Date : 1900-01-01DOI: 10.1201/9781420035315.ch19
E. Schulte
regular polytopes are combinatorial structures that generalize the familiar regular polytopes. The terminology adopted is patterned after the classical theory. Many symmetric figures discussed in earlier sections could be treated (and their structure clarified) in this more general framework. Much of the research in this area is quite recent. For a comprehensive account see McMullen and Schulte [McS02].
正多面体是将我们所熟悉的正多面体泛化而成的组合结构。所采用的术语是以经典理论为蓝本的。在前面章节中讨论的许多对称图形可以在这个更一般的框架中处理(并澄清它们的结构)。这一领域的许多研究都是最近才进行的。有关全面的说明，见McMullen and Schulte [McS02]。
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Pub Date : 1900-01-01DOI: 10.1201/9781420035315.ch51
J. O'Rourke, G. Toussaint
{"title":"Pattern recognition","authors":"J. O'Rourke, G. Toussaint","doi":"10.1201/9781420035315.ch51","DOIUrl":"https://doi.org/10.1201/9781420035315.ch51","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134415953","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 : 1900-01-01DOI: 10.1201/9781420035315.ch59
N. White
{"title":"Geometric applications of the grassmann-cayley algebra","authors":"N. White","doi":"10.1201/9781420035315.ch59","DOIUrl":"https://doi.org/10.1201/9781420035315.ch59","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"96 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134476229","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 : 1900-01-01DOI: 10.1201/9781420035315.ch12
R. Schneider
Stochastic geometry studies randomly generated geometric objects. The present chapter deals with some discrete aspects of stochastic geometry. We describe work that has been done on finite point sets, their convex hulls, infinite discrete point sets, arrangements of flats, and tessellations of space, under various assumptions of randomness. Typical results concern expectations of geometrically defined random variables, or probabilities of events defined by random geometric configurations. The selection of topics must necessarily be restrictive. We leave out the large number of special elementary geometric probability problems that can be solved explicitly by direct, though possibly intricate, analytic calculations. We pay special attention to either asymptotic results, where the number of points considered tends to infinity, or to inequalities, or to identities where the proofs involve more delicate geometric or combinatorial arguments. The close ties of discrete geometry with convexity are reflected: we consider convex hulls of random points, intersections of random halfspaces, and tessellations of space into convex sets. There are many topics that one might classify under ‘discrete aspects of stochastic geometry’, such as optimization problems with random data, the average-case analysis of geometric algorithms, random geometric graphs, random coverings, percolation, shape theory, and several others. All of these have to be excluded here.
{"title":"Discrete Aspects of Stochastic Geometry","authors":"R. Schneider","doi":"10.1201/9781420035315.ch12","DOIUrl":"https://doi.org/10.1201/9781420035315.ch12","url":null,"abstract":"Stochastic geometry studies randomly generated geometric objects. The present chapter deals with some discrete aspects of stochastic geometry. We describe work that has been done on finite point sets, their convex hulls, infinite discrete point sets, arrangements of flats, and tessellations of space, under various assumptions of randomness. Typical results concern expectations of geometrically defined random variables, or probabilities of events defined by random geometric configurations. The selection of topics must necessarily be restrictive. We leave out the large number of special elementary geometric probability problems that can be solved explicitly by direct, though possibly intricate, analytic calculations. We pay special attention to either asymptotic results, where the number of points considered tends to infinity, or to inequalities, or to identities where the proofs involve more delicate geometric or combinatorial arguments. The close ties of discrete geometry with convexity are reflected: we consider convex hulls of random points, intersections of random halfspaces, and tessellations of space into convex sets. There are many topics that one might classify under ‘discrete aspects of stochastic geometry’, such as optimization problems with random data, the average-case analysis of geometric algorithms, random geometric graphs, random coverings, percolation, shape theory, and several others. All of these have to be excluded here.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"715 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133499173","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 : 1900-01-01DOI: 10.1201/9781420035315.ch53
C. Bajaj
Piecewise polynomials of some xed degree and continuously di erentiable upto some order are known as splines or nite elements. Splines are used in applications ranging from image processing, computer aided design, to the solution of partial di erential equations via nite element analysis. The spline tting problem of constructing a mesh of nite elements that interpolate or approximate multivariate data is by far the primary research problem in geometric modeling. Parametric splines are vectors of multivariate polynomial (or rational) functions while implicit splines are zero contours of multivariate polynomials. This survey shall dwell mainly on spline surface tting methods in IR Tensor product splines in (Section xx.1,...), triangular basis splines (Section xx.7,...). The following criteria may be used in evaluating these spline methods:
{"title":"Splines and geometric modeling","authors":"C. Bajaj","doi":"10.1201/9781420035315.ch53","DOIUrl":"https://doi.org/10.1201/9781420035315.ch53","url":null,"abstract":"Piecewise polynomials of some xed degree and continuously di erentiable upto some order are known as splines or nite elements. Splines are used in applications ranging from image processing, computer aided design, to the solution of partial di erential equations via nite element analysis. The spline tting problem of constructing a mesh of nite elements that interpolate or approximate multivariate data is by far the primary research problem in geometric modeling. Parametric splines are vectors of multivariate polynomial (or rational) functions while implicit splines are zero contours of multivariate polynomials. This survey shall dwell mainly on spline surface tting methods in IR Tensor product splines in (Section xx.1,...), triangular basis splines (Section xx.7,...). The following criteria may be used in evaluating these spline methods:","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129475901","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 : 1900-01-01DOI: 10.1201/9781420035315.ch5
J. Goodman
Pseudoline arrangements generalize in a natural way arrangements of straight lines, discarding the straightness aspect, but preserving their basic topological and combinatorial properties. Elementary and intuitive in nature, at the same time, by the Folkman-Lawrence topological representation theorem (see Chapter 6), they provide a concrete geometric model for oriented matroids of rank 3. After their explicit description by Levi in the 1920’s, and the subsequent development of the theory by Ringel in the 1950’s, the major impetus was given in the 1970’s by Grünbaum’s monograph Arrangements and Spreads, in which a number of results were collected and a great many problems and conjectures posed about arrangements of both lines and pseudolines. The connection with oriented matroids discovered several years later led to further work. The theory is by now very well developed, with many combinatorial and topological results and connections to other areas as for example algebraic combinatorics, as well as a large number of applications in computational geometry. In comparison to arrangements of lines arrangements of pseudolines have the advantage that they are more general and allow for a purely combinatorial treatment. Section 5.1 is devoted to the basic properties of pseudoline arrangements, and Section 5.2 to related structures, such as arrangements of straight lines, configurations (and generalized configurations) of points, and allowable sequences of permutations. (We do not discuss the connection with oriented matroids, however; that is included in Chapter 6.) In Section 5.3 we discuss the stretchability problem. Section 5.4 summarizes some combinatorial results known about line and pseudoline arrangements, in particular problems related to the cell structure of arrangements. Section 5.5 deals with results of a topological nature and Section 5.6 with issues of combinatorial and computational complexity. Section 5.7 with several applications, including sweeping arrangements and pseudotriangulations. Unless otherwise noted, we work in the real projective plane P.
伪线排列以自然的方式推广直线排列，放弃直线性方面，但保留其基本的拓扑和组合性质。同时，通过Folkman-Lawrence拓扑表示定理(见第6章)，他们为3阶有向拟阵提供了一个具体的几何模型。20世纪20年代列维对其进行了明确的描述，50年代林格尔对这一理论进行了进一步的发展，70年代格纳鲍姆的专著《排列与扩散》(arrangement and Spreads)推动了这一理论的发展，该书收集了许多结果，提出了许多关于线和伪线排列的问题和猜想。几年后发现的与定向拟阵的联系导致了进一步的研究。这个理论现在已经发展得很好，有许多组合和拓扑结果，并与其他领域有联系，例如代数组合学，以及在计算几何中的大量应用。与线的排列相比，伪线的排列的优点是它们更一般，并且允许进行纯粹的组合处理。第5.1节专门讨论伪线排列的基本性质，第5.2节讨论相关结构，如直线排列、点的构型(和广义构型)以及允许的排列序列。(然而，我们不讨论与取向拟阵的联系;这是第6章的内容。)在5.3节中，我们将讨论可拉伸性问题。第5.4节总结了一些已知的关于线和伪线排列的组合结果，特别是与排列的细胞结构有关的问题。第5.5节讨论拓扑性质的结果，第5.6节讨论组合和计算复杂性的问题。5.7节给出了几个应用，包括扫描排列和伪三角测量。除非特别说明，我们在实投影平面P上工作。
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Pub Date : 1900-01-01DOI: 10.1201/9781420035315.ch62
M. Senechal
{"title":"Crystals and quasicrystals","authors":"M. Senechal","doi":"10.1201/9781420035315.ch62","DOIUrl":"https://doi.org/10.1201/9781420035315.ch62","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129468872","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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