Contents
Chapter 1. Introduction by Morten Dæhlen
1
Chapter 2. Triangulation Preliminaries
2.1. Notation and Terminology
2.2. Properties of Triangulations
2.3. A Simple Triangulation Algorithm
2.4. Exercises
3
3
5
7
10
Chapter 3. Graphs and Data Structures
3.1. Graph Theoretic Concepts
3.2. Generalized Maps (G-maps)
3.3. Data Structures for Triangulations
3.4. A Minimal Triangle Based Data Structure
3.5. Triangle-Based Data Structure with Neighbours
3.6. Vertex-Based Data Structure with Neighbours
3.7. Half-Edge Data Structure
3.8. Dart-Based Data Structure
3.9. Notes on Object-Oriented Design and Implementation
3.10. Exercises
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11
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Chapter 4. Delaunay Triangulations and Voronoi Diagrams
4.1. Optimal Triangulations
4.2. Voronoi diagrams
4.3. Delaunay Triangulation Obtained from the Voronoi Diagram
4.4. The Circle Criterion
4.5. Equivalence of the Delaunay Criteria for Strictly Convex Quadrilaterals
4.6. Computing the Circumcircle Test
4.7. The Local Optimization Procedure (LOP)
4.8. Global Properties of the Delaunay Triangulation
4.9. Exercises
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Chapter 5. Algorithms for Delaunay Triangulation
5.1. A Simple Algorithm Based on Previous Results
5.2. Radial Sweep
5.3. A Step-by-Step Approach for Making Delaunay Triangles
5.4. Incremental Algorithms
5.5. Inserting a Point into a Delaunay Triangulation
5.6. Point Insertion and Edge Swapping
5.7. Running Time of Incremental Algorithms
5.8. Point Location in a Triangulation
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CONTENTS
5.9. Divide-and-Conquer
5.10. Exercises
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Chapter 6. Data Dependent Triangulations
6.1. Motivation
6.2. Optimal Triangulations Revisited
6.3. The General Concept
6.4. Data Dependent Swapping Criteria
6.5. On Implementation of the LOP
6.6. Modified Local Optimization Schemes (MLOP)
6.7. Simulated Annealing
6.8. Exercises
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Chapter 7. Constrained Delaunay Triangulation
7.1. Delaunay Triangulation of a Planar Straight-Line Graph
7.2. Generalization of Delaunay Triangulation
7.3. Algorithms for Constrained Delaunay Triangulation
7.4. Inserting an Edge into a CDT
7.5. Edge Insertion and Swapping
7.6. Inserting a Point into a CDT
7.7. Exercises
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Chapter 8. Delaunay Refinement Mesh Generation1
8.1. Introduction
8.2. General Requirements for Meshes
8.3. Node Insertion
8.4. Splitting Encroached Segments
8.5. The Delaunay Refinement Algorithm
8.6. Minimum Edge Length and Termination
8.7. Corner-Lopping for Handling Small Input Angles
8.8. Spatial Grading
8.9. Exercises
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Chapter 9. Programming Triangulations:
The Triangulation Template Library (TTL)
9.1. Implementing The Half-Edge Data Structure
9.2. The Overall Design and the Adaptation Layer
9.3. Topological Queries and the Dart Class
9.4. Some Iterator Classes
9.5. Geometric Queries and the Traits Class
9.6. Geometric and Topological Modifiers
9.7. Generic Delaunay Triangulation
9.8. Conclusion
9.9. Exercises
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Chapter 10. Least Squares Approximation of Scattered Data over
Triangulations
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1
Some of the illustrations in this issue of the chapter are taken from papers published in
journals. It is explicitly mentioned in the figures’ captions when this is done. Other illustrations
are in progress for later issues.
CONTENTS
10.1.
10.2.
10.3.
10.4.
10.5.
10.6.
10.7.
10.8.
Another formulation of surface triangulations
Approximation on triangulations of subsets of data
Existence and Uniqueness
Sparsity and Symmetry
Penalized Least Squares
Smoothing Terms for Penalized Least Squares
Approximation over General Triangulatons
Exercises
Bibliography
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