1651887551

Space-Filling Curves : An Introduction with Applications in Scientific Computing / by Michael Bader

Englisch

Berlin ; Heidelberg : Springer, 2013

Online-Ressource (XIII, 278 p. 357 illus., 323 illus. in color, digital)

Description based upon print version of record

Buchausg. u.d.T.: Bader, Michael *1971-*: Space-filling curves. - Heidelberg : Springer, 2013. - xiii, 278 Seiten

978-3-642-31046-1

Weitere Ausgaben: 978-3-642-31045-4 (Druckausgabe)

DOI: 10.1007/978-3-642-31046-1

""Space-Filling Curves""; ""Preface""; ""How to Read This Book""; ""Acknowledgements""; ""Contents""; ""Chapter 1 Two Motivating Examples: Sequential Orders on Quadtrees and Multidimensional Data Structures ""; ""1.1 Modelling Complicated Geometries with Quadtrees, Octrees, and Spacetrees""; ""1.1.1 Quadtrees and Octrees""; ""1.1.2 A Sequential Order on Quadtree Cells""; ""1.1.3 A More Local Sequential Order on Quadtree Cells""; ""1.2 Numerical Simulation: Solving a Simple Heat Equation""; ""1.3 Sequentialisation of Multidimensional Data""; ""1.3.1 Requirements for Efficient Sequential Orders""""1.3.2 Row-Major and Column-Major Sequentialisation""; ""Chapter 2 How to Construct Space-Filling Curves ""; ""2.1 Towards a Bijective Mapping of the Unit Interval to the Unit Square""; ""2.2 Continuous Mappings and (Space-Filling) Curves""; ""2.3 The Hilbert Curve""; ""2.3.1 Iterations of the Hilbert Curve""; ""2.3.2 Approximating Polygons""; ""2.3.3 Definition of the Hilbert Curve""; ""2.3.4 Proof: h Defines a Space-Filling Curve""; ""2.3.5 Continuity of the Hilbert Curve""; ""2.3.6 Moore's Version of the Hilbert Curve""; ""2.4 Peano Curve""""2.5 Space-Filling Curves: Required Algorithms""; ""Chapter 3 Grammar-Based Description of Space-Filling Curves ""; ""3.1 Description of the Hilbert Curve Using Grammars""; ""3.2 A Traversal Algorithm for 2D Data""; ""3.3 Grammar-Based Description of the Peano Curve""; ""3.4 A Grammar for Turtle Graphics""; ""Chapter 4 Arithmetic Representation of Space-Filling Curves ""; ""4.1 Arithmetic Representation of the Hilbert Mapping""; ""4.2 Calculating the Values of h""; ""4.3 Uniqueness of the Hilbert Mapping""; ""4.4 Computation of the Inverse: Hilbert Indices"" ; ""4.5 Arithmetisation of the Peano Curve""""4.6 Efficient Computation of Space-Filling Mappings""; ""4.6.1 Computing Hilbert Mappings via Recursion Unrolling ""; ""4.6.2 From Recursion Unrolling to State Diagrams""; ""Chapter 5 Approximating Polygons ""; ""5.1 Approximating Polygons of the Hilbert and Peano Curve""; ""5.2 Measuring Curve Lengths with Approximating Polygons""; ""5.3 Fractal Curves and Their Length""; ""5.4 A Quick Excursion on Fractal Curves""; ""Chapter 6 Sierpinski Curves ""; ""6.1 The Sierpinski-Knopp Curve""; ""6.1.1 Construction of the Sierpinski Curve"";""6.1.2 Grammar-Based Description of the Sierpinski Curve""""6.1.3 Arithmetisation of the Sierpinski Curve""; ""6.1.4 Computation of the Sierpinski Mapping""; ""6.2 Generalised Sierpinski Curves""; ""6.2.1 Bisecting Triangles Along Tagged Edges""; ""6.2.2 Continuity and Locality of Generalised Sierpinski Curves""; ""6.2.3 Filling Triangles with Curved Edges""; ""Chapter 7 Further Space-Filling Curves ""; ""7.1 Characterisation of Space-Filling Curves""; ""7.2 Lebesgue Curve and Morton Order""; ""7.3 The H-Index""; ""7.4 The βΩ-Curve""; ""7.5 The Gosper Flowsnake"";""Chapter 8 Space-Filling Curves in 3D ""

*Raumfüllende Kurve

*Raumfüllende Kurve

*Computer science ; Mathematics ; Algorithms ; Computer mathematics. ; Applied mathematics. ; Engineering mathematics. ; Computer science—Mathematics. ; Mathematics ; Computer science ; Algorithms

*Curves / Mathematical models ; Curves on surfaces / Mathematical models

Klassifikation der Library of Congress: QA71-90

Dewey Dezimal-Klassifikation: 004;

Regensburger Verbund-Klassifikation: SK 280: SA-SP Mathematik / SK Monografien / Topologie allgemein Allgemeine Lehrbücher

bisacsh: MAT003000

Book Industry Communication: PDE

bisacsh: COM014000

Mathematics Subject Classification: *68-02

Mathematics Subject Classification: 68U05

Mathematics Subject Classification: 54F50

Mathematics Subject Classification: 28A80

Mathematics Subject Classification: 65D17

Mathematics Subject Classification: 65Y05

Mathematics Subject Classification: 68W25

Two Motivating Examples -- How to Construct Space-Filling Curves -- Grammar-Based Description of Space-Filling Curves -- Arithmetic Representation of Space-Filling Curves -- Approximating Polygons -- Sierpinski Curves -- Further Space-Filling Curves -- Space-Filling Curves in 3D -- Refinement Trees and Space-Filling Curves -- Parallelisation with Space-Filling Curves -- Locality Properties of Space-Filling Curves -- Sierpinski Curves on Triangular and Tetrahedral Meshes -- Case Study: Cache Efficient Algorithms for Matrix Operations -- Case Study: Numerical Simulation on Spacetree Grids Using Space-Filling Curves.- Further Applications of Space-Filling Curves.- Solutions to Selected Exercises.- References -- Index .

­The present book provides an introduction to using space-filling curves (SFC) as tools in scientific computing. Special focus is laid on the representation of SFC and on resulting algorithms. For example, grammar-based techniques are introduced for traversals of Cartesian and octree-type meshes, and arithmetisation of SFC is explained to compute SFC mappings and indexings. ­The locality properties of SFC are discussed in detail, together with their importance for algorithms. Templates for parallelisation and cache-efficient algorithms are presented to reflect the most important applications of SFC in scientific computing. Special attention is also given to the interplay of adaptive mesh refinement and SFC, including the structured refinement of triangular and tetrahedral grids. For each topic, a short overview is given on the most important publications and recent research activities.

https://zbmath.org/?q=an:1283.68012 [Zentralblatt MATH]

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https://doi.org/10.1007/978-3-642-31046-1 [E-books (Springer)]