Pub Date : 1900-01-01DOI: 10.1201/9781420035315.ch21
E. Schulte, U. Brehm
{"title":"Polyhedral Maps","authors":"E. Schulte, U. Brehm","doi":"10.1201/9781420035315.ch21","DOIUrl":"https://doi.org/10.1201/9781420035315.ch21","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"106 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":"122964381","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.ch49
D. Dobkin, S. Teller
Computer animation is the use of computers to create animations. There are a few different ways to make computer animations. One is 3D animation. One way to create computer animations is to create objects and then render them. This method produces perfect and three dimensional looking animations. Another way to create computer animation is to use standard computer painting tools and to paint single frames and composite them. These can later be either saved as a movie file or output to video. One last method of making computer animations is to use transitions and other special effects like morphing to modify existing images and video. Computer graphics are any types of images created using any kind of computer. There is a vast amount of types of images a computer can create. Also, there are just as many ways of creating those images. Images created by computers can be very simple, such as lines and circles, or extremly complex such as fractals and complicated rendered animations. If you want to create your own computer graphics, no matter how simple or complex, you have to know a few things about computers, computer graphics, and how they work. The following information should help you get started in the field of computer graphics:
{"title":"Computer graphics","authors":"D. Dobkin, S. Teller","doi":"10.1201/9781420035315.ch49","DOIUrl":"https://doi.org/10.1201/9781420035315.ch49","url":null,"abstract":"Computer animation is the use of computers to create animations. There are a few different ways to make computer animations. One is 3D animation. One way to create computer animations is to create objects and then render them. This method produces perfect and three dimensional looking animations. Another way to create computer animation is to use standard computer painting tools and to paint single frames and composite them. These can later be either saved as a movie file or output to video. One last method of making computer animations is to use transitions and other special effects like morphing to modify existing images and video. Computer graphics are any types of images created using any kind of computer. There is a vast amount of types of images a computer can create. Also, there are just as many ways of creating those images. Images created by computers can be very simple, such as lines and circles, or extremly complex such as fractals and complicated rendered animations. If you want to create your own computer graphics, no matter how simple or complex, you have to know a few things about computers, computer graphics, and how they work. The following information should help you get started in the field of computer graphics:","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":"130813636","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.ch13
J. Alexander, J. Beck, William W. L. Chen
A sequence s1, s2, . . . in U = [0, 1) is said to be uniformly distributed if, in the limit, the number of sj falling in any given subinterval is proportional to its length. Equivalently, s1, s2, . . . is uniformly distributed if the sequence of equiweighted atomic probability measures μN (sj) = 1/N , supported by the initial N -segments s1, s2, . . . , sN , converges weakly to Lebesgue measure on U. This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets. Uniform distribution, as an area of study, originated from the remarkable paper of Weyl [Wey16], in which he established the fundamental result known nowadays as the Weyl criterion (see [Cas57, KN74]). This reduces a problem on uniform distribution to a study of related exponential sums, and provides a deeper understanding of certain aspects of Diophantine approximation, especially basic results such as Kronecker’s density theorem. Indeed, careful analysis of the exponential sums that arise often leads to Erdős-Turán type upper bounds, which in turn lead to quantitative statements concerning uniform distribution. Today, the concept of uniform distribution has important applications in a number of branches of mathematics such as number theory (especially Diophantine approximation), combinatorics, ergodic theory, discrete geometry, statistics, numerical analysis, etc. In this chapter, we focus on the geometric aspects of the theory.
{"title":"Geometric Discrepancy Theory Anduniform Distribution","authors":"J. Alexander, J. Beck, William W. L. Chen","doi":"10.1201/9781420035315.ch13","DOIUrl":"https://doi.org/10.1201/9781420035315.ch13","url":null,"abstract":"A sequence s1, s2, . . . in U = [0, 1) is said to be uniformly distributed if, in the limit, the number of sj falling in any given subinterval is proportional to its length. Equivalently, s1, s2, . . . is uniformly distributed if the sequence of equiweighted atomic probability measures μN (sj) = 1/N , supported by the initial N -segments s1, s2, . . . , sN , converges weakly to Lebesgue measure on U. This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets. Uniform distribution, as an area of study, originated from the remarkable paper of Weyl [Wey16], in which he established the fundamental result known nowadays as the Weyl criterion (see [Cas57, KN74]). This reduces a problem on uniform distribution to a study of related exponential sums, and provides a deeper understanding of certain aspects of Diophantine approximation, especially basic results such as Kronecker’s density theorem. Indeed, careful analysis of the exponential sums that arise often leads to Erdős-Turán type upper bounds, which in turn lead to quantitative statements concerning uniform distribution. Today, the concept of uniform distribution has important applications in a number of branches of mathematics such as number theory (especially Diophantine approximation), combinatorics, ergodic theory, discrete geometry, statistics, numerical analysis, etc. In this chapter, we focus on the geometric aspects of the theory.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"17 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":"131869469","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.ch33
B. Mishra
Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer aided design, geometric theorem proving, etc. The following survey paper discusses the underlying concepts, algorithms and a series of representative applications. This paper will appear as a chapter in the "Handbook of Discrete and Computational Geometry" (Edited by J.E. Goodman and J. O''Rourke), CRC Series in Discrete and Combinatorial Mathematics.
{"title":"Computational Real Algebraic Geometry","authors":"B. Mishra","doi":"10.1201/9781420035315.ch33","DOIUrl":"https://doi.org/10.1201/9781420035315.ch33","url":null,"abstract":"Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer aided design, geometric theorem proving, etc. The following survey paper discusses the underlying concepts, algorithms and a series of representative applications. This paper will appear as a chapter in the \"Handbook of Discrete and Computational Geometry\" (Edited by J.E. Goodman and J. O''Rourke), CRC Series in Discrete and Combinatorial Mathematics.","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"50 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":"127777032","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.ch61
G. Kabatiansky, J. A. Rush
{"title":"Sphere packing and coding theory","authors":"G. Kabatiansky, J. A. Rush","doi":"10.1201/9781420035315.ch61","DOIUrl":"https://doi.org/10.1201/9781420035315.ch61","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"37 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":"115514649","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.ch52
R. Tamassia, G. Liotta
{"title":"Graph drawing","authors":"R. Tamassia, G. Liotta","doi":"10.1201/9781420035315.ch52","DOIUrl":"https://doi.org/10.1201/9781420035315.ch52","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"15 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":"122654933","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.ch20
G. Kalai
{"title":"Polytope Skeletons and Paths","authors":"G. Kalai","doi":"10.1201/9781420035315.ch20","DOIUrl":"https://doi.org/10.1201/9781420035315.ch20","url":null,"abstract":"","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"90 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":"122725390","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.pt2
M. Henk, Jürgen Richter-Gebert, G. Ziegler
Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, provide a sketch of “what polytopes look like” and “how they behave,” with many explicit examples, and briefly state some main results (where further details are in the subsequent chapters of this Handbook). We concentrate on two main topics:
{"title":"Basic Properties of Convex Polytopes","authors":"M. Henk, Jürgen Richter-Gebert, G. Ziegler","doi":"10.1201/9781420035315.pt2","DOIUrl":"https://doi.org/10.1201/9781420035315.pt2","url":null,"abstract":"Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, provide a sketch of “what polytopes look like” and “how they behave,” with many explicit examples, and briefly state some main results (where further details are in the subsequent chapters of this Handbook). We concentrate on two main topics:","PeriodicalId":156768,"journal":{"name":"Handbook of Discrete and Computational Geometry, 2nd Ed.","volume":"1994 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":"124989756","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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