Handbook of Discrete and Computational Geometry, 2nd Ed.最新文献_第4页

Symmetry of Polytopes and Polyhedra 多面体和多面体的对称性

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch19

E. Schulte

引用次数: 14

Pattern recognition 模式识别

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch51

J. O'Rourke, G. Toussaint

引用次数: 1

Geometric applications of the grassmann-cayley algebra 格拉斯曼-凯利代数的几何应用

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch59

N. White

引用次数: 20

Discrete Aspects of Stochastic Geometry 随机几何的离散方面

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch12

R. Schneider

{"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}

引用次数: 58

Splines and geometric modeling 样条和几何建模

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch53

C. Bajaj

引用次数: 3

Modeling motion 运动建模

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch50

L. Guibas

引用次数: 2

Convex Hull Computations 凸壳计算

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.pt3

R. Seidel

引用次数: 73

Pseudoline Arrangements Pseudoline安排

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch5

J. Goodman

{"title":"Pseudoline Arrangements","authors":"J. Goodman","doi":"10.1201/9781420035315.ch5","DOIUrl":"https://doi.org/10.1201/9781420035315.ch5","url":null,"abstract":"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.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"613 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":"122940942","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}

引用次数: 80

Crystals and quasicrystals 晶体和准晶体

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.ch62

M. Senechal

引用次数: 5

Point Location 点位置

Handbook of Discrete and Computational Geometry, 2nd Ed. Pub Date : 1900-01-01 DOI: 10.1201/9781420035315.pt4

J. Snoeyink

引用次数: 33