Introductory Tiling Theory for Computer Graphics (Synthesis Lectures on Computer Graphics and Animation)

✍ Scribed by Craig Kaplan

• Publisher: Morgan and Claypool Publishers
• Year: 2009
• Tongue: English
• Leaves: 113
• Series: Synthesis Lectures on Computer Graphics and Animation
• Edition: 1
• Category: Library

✦ Synopsis

Tiling theory is an elegant branch of mathematics that has applications in several areas of computer science. The most immediate application area is graphics, where tiling theory has been used in the contexts of texture generation, sampling theory, remeshing, and of course the generation of decorative patterns. The combination of a solid theoretical base (complete with tantalizing open problems), practical algorithmic techniques, and exciting applications make tiling theory a worthwhile area of study for practitioners and students in computer science. This synthesis lecture introduces the mathematical and algorithmic foundations of tiling theory to a computer graphics audience. The goal is primarily to introduce concepts and terminology, clear up common misconceptions, and state and apply important results. The book also describes some of the algorithms and data structures that allow several aspects of tiling theory to be used in practice. Table of Contents: Introduction / Tiling Basics / Symmetry / Tilings by Polygons / Isohedral Tilings / Nonperiodic and Aperiodic Tilings / Survey

✦ Table of Contents

Preface......Page 9
Organization......Page 11
Defining tilings......Page 13
Anatomy of a tiling......Page 15
Tilings with congruent tiles......Page 16
The set of symmetries......Page 21
Symmetry groups......Page 22
Factoring out repetition......Page 27
Periodic replication......Page 28
Symmetries of tilings......Page 30
Colour symmetry......Page 32
Orbifolds......Page 34
Regular and uniform tilings......Page 39
Laves tilings......Page 40
Basic definitions......Page 45
Isohedral tiling types......Page 47
Edge shape parameterization......Page 50
Tiling vertex parameterization......Page 51
Representing tiling vertex parameterizations......Page 54
Computing transformation matrices......Page 55
Colourings......Page 57
Isohedral templates and prototiles......Page 58
Beyond isohedral tilings......Page 60
Nonperiodic and Aperiodic Tilings......Page 65
Substitution tilings and rep-tiles......Page 67
Wang tiles and Aperiodicity......Page 72
Penrose tilings......Page 74
Escher-like tilings......Page 81
Sampling......Page 82
Texture generation......Page 83
The Isohedral Tiling Types......Page 85
Bibliography......Page 109
Biography......Page 113