Delaunay triangulation

Overview

Delaunay triangulation is a fundamental geometric structure in computational geometry, named after the Russian mathematician Boris Delaunay. It is a triangulation of a set of points in a plane such that no point is inside the circumcircle of any triangle in the triangulation. This property makes Delaunay triangulations useful in various applications, including computer graphics, geographic information systems (GIS), and numerical simulations.

Definition and Properties

Delaunay triangulation for a set of points P in the plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). This definition ensures that the triangles are as "equilateral" as possible, avoiding skinny triangles. The Delaunay triangulation maximizes the minimum angle of all the angles of the triangles in the triangulation, which is known as the max-min angle property.

Circumcircle Property

A circumcircle of a triangle is a circle that passes through all three vertices of the triangle. The Delaunay triangulation has the property that for each triangle in the triangulation, the circumcircle of that triangle does not contain any other points from the set P inside it. This property is crucial for ensuring that the triangles are well-shaped and not elongated.

Empty Circle Property

The empty circle property states that for any edge in the Delaunay triangulation, there exists a circle passing through the endpoints of the edge that does not contain any other points from the set P. This property is a direct consequence of the circumcircle property and is used in many algorithms for constructing Delaunay triangulations.

Algorithms for Construction

Several algorithms exist for constructing Delaunay triangulations, each with different computational complexities and practical performance characteristics. Some of the most well-known algorithms include:

Incremental Algorithm

The incremental algorithm constructs the Delaunay triangulation by adding one point at a time and updating the triangulation accordingly. This algorithm has a worst-case time complexity of O(n^2), but it can be optimized to O(n log n) on average by using efficient data structures.

Divide and Conquer Algorithm

The divide and conquer algorithm splits the set of points into smaller subsets, constructs the Delaunay triangulation for each subset, and then merges the triangulations. This algorithm has a time complexity of O(n log n) and is efficient for large datasets.

Sweep Line Algorithm

The sweep line algorithm constructs the Delaunay triangulation by sweeping a line across the plane and maintaining the triangulation dynamically. This algorithm also has a time complexity of O(n log n) and is suitable for real-time applications.

Bowyer-Watson Algorithm

The Bowyer-Watson algorithm is an incremental algorithm that adds points one at a time and removes triangles whose circumcircles contain the new point. New triangles are then formed by connecting the new point to the vertices of the removed triangles. This algorithm is simple to implement and has a time complexity of O(n log n) on average.

Applications

Delaunay triangulations have numerous applications in various fields due to their desirable properties.

Computer Graphics

In computer graphics, Delaunay triangulations are used for mesh generation, surface reconstruction, and texture mapping. The well-shaped triangles produced by Delaunay triangulation are ideal for rendering and simulation purposes.

Geographic Information Systems (GIS)

In GIS, Delaunay triangulations are used for terrain modeling, spatial analysis, and interpolation. The triangulation provides a natural way to represent the surface of the Earth and perform various spatial operations.

Numerical Simulations

Delaunay triangulations are used in numerical simulations for finite element analysis, fluid dynamics, and structural analysis. The max-min angle property ensures that the elements are well-shaped, leading to more accurate and stable simulations.

Mathematical Background

The mathematical foundation of Delaunay triangulation involves several key concepts from computational geometry and graph theory.

Voronoi Diagrams

The Voronoi diagram is a partitioning of the plane into regions based on the distance to a specific set of points. Each region contains all the points closer to one particular point than to any other. The Delaunay triangulation is the dual graph of the Voronoi diagram, meaning that each vertex of the Delaunay triangulation corresponds to a region in the Voronoi diagram.

Gabriel Graphs

The Gabriel graph is a subgraph of the Delaunay triangulation where an edge exists between two points if and only if the circle with the edge as its diameter does not contain any other points from the set. The Gabriel graph is used in various applications, including wireless network design and clustering.

Empty Sphere Property in Higher Dimensions

In higher dimensions, the Delaunay triangulation generalizes to the concept of an empty sphere. For a set of points in three-dimensional space, the Delaunay triangulation consists of tetrahedra such that no point is inside the circumsphere of any tetrahedron. This property is essential for applications in three-dimensional modeling and simulations.

Computational Complexity

The computational complexity of constructing Delaunay triangulations depends on the algorithm used and the dimensionality of the space.

Planar Delaunay Triangulation

For a set of n points in the plane, the Delaunay triangulation can be constructed in O(n log n) time using efficient algorithms such as the divide and conquer or sweep line algorithms. The space complexity is O(n) since the triangulation consists of O(n) triangles.

Higher-Dimensional Delaunay Triangulation

In higher dimensions, the complexity of constructing Delaunay triangulations increases significantly. For a set of n points in d-dimensional space, the worst-case time complexity is O(n^(ceil(d/2))). However, practical algorithms often perform better than the worst-case analysis suggests.

Robustness and Numerical Stability

Constructing Delaunay triangulations in practice requires careful consideration of numerical stability and robustness.

Floating-Point Arithmetic

Using floating-point arithmetic can lead to numerical errors and inconsistencies in the triangulation. Robust algorithms often use exact arithmetic or adaptive precision techniques to ensure the correctness of the triangulation.

Degenerate Cases

Degenerate cases, such as collinear points or points on a common circle, can cause difficulties in constructing Delaunay triangulations. Special handling of these cases is necessary to ensure the robustness of the algorithm.

Extensions and Variants

Several extensions and variants of Delaunay triangulation have been developed to address specific needs and applications.

Constrained Delaunay Triangulation

Constrained Delaunay triangulation (CDT) is a variant where certain edges are required to be part of the triangulation. This is useful in applications where the triangulation must conform to a given set of constraints, such as in mesh generation for finite element analysis.

Weighted Delaunay Triangulation

Weighted Delaunay triangulation assigns weights to the points, and the triangulation is constructed based on the weighted distances. This variant is used in applications where different points have varying importance or influence.

Higher-Order Delaunay Triangulation

Higher-order Delaunay triangulations consider not just the immediate neighbors but also second or higher-order neighbors. These triangulations are used in applications requiring more complex connectivity and relationships between points.

References