Format:
Online-Ressource (X, 539p. 496 illus., 281 illus. in color, digital)
ISBN:
9783642129711
,
1280391774
,
9781280391774
Series Statement:
Algorithms and Computation in Mathematics 25
Content:
Triangulations in Mathematics -- Configurations, Triangulations, Subdivisions, and Flips -- Life in Two Dimensions -- A Tool Box -- Regular Triangulations and Secondary Polytopes -- Some Interesting Configurations -- Some Interesting Triangulations -- Algorithmic Issues -- Further Topics.
Content:
Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents the first comprehensive treatment of the theory of secondary polytopes and related topics. A central theme of the book is the use of the rich structure of the space of triangulations to solve computational problems (e.g., counting the number of triangulations or finding optimal triangulations with respect to various criteria), and to establish connections to applications in algebra, computer science, combinatorics, and optimization. With many examples and exercises, and with nearly five hundred illustrations, the book gently guides readers through the properties of the spaces of triangulations of "structured" (e.g., cubes, cyclic polytopes, lattice polytopes) and "pathological" (e.g., disconnected spaces of triangulations) situations using only elementary principles.
Note:
Includes bibliographical references and index
,
""Triangulations ""; ""Preface""; ""Contents ""; ""1 Triangulations in Mathematics""; ""1.1 Combinatorics and triangulations""; ""1.2 Optimization and triangulations""; ""1.3 Algebra and triangulations""; ""1.4 The rest of this book""; "" Exercises""; ""2 Configurations, Triangulations, Subdivisions, and Flips""; ""2.1 The official languages in the land of triangulations""; ""2.1.1 Polyhedra and cones""; ""2.1.2 Point configurations""; ""2.1.3 Geometry of point configurations""; ""2.2 A closer look at the definition of triangulation""; ""2.2.1 There is always a triangulation""
,
""2.2.2 A famous example: the Delaunay triangulation""""2.2.3 Regular subdivisions and their structure""; ""2.3 A bullet-proof definition of polyhedral subdivisions""; ""2.3.1 Polyhedral subdivisions""; ""2.3.2 Regular subdivisions, again""; ""2.4 Flips and the graph of triangulations""; ""2.4.1 Corank-one configurations and circuits""; ""2.4.2 Almost-triangulations and flips""; ""2.5 Vector configurations and their triangulations""; ""2.5.1 Vector configurations""; ""2.5.2 Polyhedral subdivisions of vector configurations""; ""2.5.3 Regular subdivisions of vector configurations""
,
""2.6 Triangulations as simplicial complexes""""2.6.1 Simplicial complexes""; ""2.6.2 The f-vector of a simplicial complexes""; ""2.6.3 Linear constraints on the f-vector""; "" Exercises""; ""3 Life in Two Dimensions""; ""3.1 Some basic properties""; ""3.2 A few examples of triangulations in the plane""; ""3.2.1 Placing and pulling triangulations""; ""3.2.2 Delaunay triangulations""; ""3.2.3 Greedy and minimum weight triangulations""; ""3.3 The set of all triangulations of a point set""; ""3.3.1 The exact number of triangulations""; ""3.3.2 The maximum possible number of triangulations""
,
""3.3.3 The minimum possible number of triangulations""""3.3.4 The poset of subdivisions""; ""3.4 Flips in triangulations""; ""3.4.1 All triangulations of a point set in the plane are connected by flips""; ""3.4.2 Effective enumeration of triangulations""; ""3.4.3 Further properties of the graph of flips""; ""3.5 Pseudo-triangulations""; ""3.6 Life in three dimensions""; ""3.6.1 The number of tetrahedra""; ""3.6.2 Monotone flipping does not (always) work""; ""3.6.3 The number of flips""; ""3.7 Notes and References""; "" Exercises""; ""4 A Tool Box""; ""4.1 Combinatorics of configurations""
,
""4.1.1 Dependences, circuits, and the intersection property""""4.1.2 Evaluations, cocircuits, and the union property""; ""4.1.3 Gale transforms and the duality between circuits and cocircuits""; ""4.2 Manipulating vector configurations""; ""4.2.1 Pyramids and joins""; ""4.2.2 Prisms and products""; ""4.2.3 Deletion""; ""4.2.4 Contraction""; ""4.2.5 One-point suspension""; ""4.3 Generating polyhedral subdivisions""; ""4.3.1 The placing (or pushing) triangulation""; ""4.3.2 The pulling triangulation""; ""4.3.3 Lexicographic triangulations""; ""4.3.4 Pushing and pulling refinements""
,
""4.4 Two equivalent characterizations of flips""
Additional Edition:
ISBN 9783642129704
Additional Edition:
Buchausg. u.d.T. De Loera, Jesús A., 1966 - Triangulations Berlin, Heidelberg : Springer, 2010 ISBN 3642129706
Additional Edition:
ISBN 9783642129704
Language:
English
Subjects:
Mathematics
Keywords:
Triangulation
;
Triangulation
DOI:
10.1007/978-3-642-12971-1
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
De Loera, Jesús A. 1966-
Author information:
Rambau, Jörg 1966-
