Kaplan C.S. Introductory Tiling Theory for Computer Graphics

Morgan and Claypool Publishers, 2009. — 116 p. — (Synthesis Lectures in Computer Graphics and Animation) — ISBN: 9781608450176Tiling 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.Organization
Tiling Basics
Defining tilings
Anatomy of a tiling
Patches
Tilings with congruent tiles
Symmetry
The set of symmetries
Symmetry groups
Factoring out repetition
Periodic replication
Symmetries of tilings
Other forms of symmetry
Tilings by Polygons
Regular and uniform tilings
Laves tilings
Isohedral Tilings
Basic definitions
Isohedral tiling types
Parameterizing the isohedral tilings
Data structures and algorithms for IH
Beyond isohedral tilings
Nonperiodic and Aperiodic Tilings
Substitution tilings and rep-tiles
Wang tiles and Aperiodicity
Penrose tilings
Survey
Drawing periodic tilings
Drawing nonperiodic tilings
Escher-like tilings
Sampling
Texture generation
A The Isohedral Tiling Types