📐Computational Geometry Unit 4 – Voronoi Diagrams & Delaunay Triangulations

What's the Big Idea?

• Voronoi diagrams partition a plane into regions based on the closest point from a given set of points
• Each region, called a Voronoi cell, contains all points closer to its associated point than to any other point in the set
• Delaunay triangulations are the dual graph of Voronoi diagrams, connecting points whose Voronoi cells share an edge
• Voronoi diagrams and Delaunay triangulations have wide-ranging applications in computational geometry, computer graphics, and spatial analysis
• Understanding the properties and construction of these structures is crucial for efficient problem-solving in various domains
• Voronoi diagrams provide a natural way to partition space based on proximity to a set of objects or sites
• Delaunay triangulations offer a well-defined and optimal triangulation of a set of points in the plane

Key Concepts and Definitions

• Voronoi diagram: a partition of a plane into regions based on the closest point from a given set of points
• Voronoi cell: a region in a Voronoi diagram containing all points closer to its associated point than to any other point in the set
  • Also known as a Voronoi region or Voronoi polygon
• Voronoi edge: a line segment shared by two adjacent Voronoi cells
• Voronoi vertex: a point where three or more Voronoi edges meet
• Delaunay triangulation: the dual graph of a Voronoi diagram, connecting points whose Voronoi cells share an edge
  • Satisfies the empty circle property: no point in the set is inside the circumcircle of any triangle in the triangulation
• Circumcircle: a circle that passes through all three vertices of a triangle
• Convex hull: the smallest convex polygon that contains all points in a set

Historical Background

• Voronoi diagrams are named after Georgy Voronoy, a Russian mathematician who defined and studied them in 1908
  • Voronoy's work generalized earlier concepts by Dirichlet and Descartes
• Delaunay triangulations are named after Boris Delaunay, a Russian mathematician who introduced them in 1934
• Voronoi diagrams and Delaunay triangulations have been rediscovered and applied in various fields over the years
  • Computational geometry, computer graphics, geographic information systems (GIS), and more
• The development of efficient algorithms for constructing these structures has been a major focus of research
  • Fortune's algorithm, incremental insertion, divide-and-conquer, and sweep line algorithms
• The study of Voronoi diagrams and Delaunay triangulations has led to numerous extensions and generalizations
  • Higher dimensions, weighted points, constrained triangulations, and more

Constructing Voronoi Diagrams

• Naive approach: for each point, compute the intersection of half-planes defined by the perpendicular bisectors with all other points
  • Time complexity: $O(n^3)$ for $n$ points
• Incremental insertion: add points one by one, updating the Voronoi diagram at each step
  • Time complexity: $O(n^2)$ for $n$ points
• Divide-and-conquer: recursively partition the point set, construct Voronoi diagrams for subsets, and merge them
  • Time complexity: $O(n \log n)$ for $n$ points
• Fortune's algorithm: a sweep line algorithm that maintains a beach line and a priority queue of events
  • Time complexity: $O(n \log n)$ for $n$ points
  • Considered the most efficient algorithm for constructing Voronoi diagrams
• Delaunay triangulation: construct the Delaunay triangulation first, then derive the Voronoi diagram as its dual graph
  • Various algorithms available, such as incremental insertion or divide-and-conquer

Properties of Voronoi Diagrams

• Each Voronoi cell is a convex polygon
• Voronoi edges are perpendicular bisectors of the line segments connecting the associated points
• Voronoi vertices are equidistant from three or more points in the set
• The number of Voronoi cells is equal to the number of points in the set
• The number of Voronoi edges and vertices is linear in the number of points
  • Euler's formula: $v - e + f = 2$, where $v$ is the number of vertices, $e$ is the number of edges, and $f$ is the number of faces (including the unbounded face)
• The Voronoi diagram of a set of points is unique
• The Voronoi diagram of a set of points in the plane has a total complexity of $O(n)$, where $n$ is the number of points

Delaunay Triangulations: The Dual Graph

• Delaunay triangulation is the dual graph of the Voronoi diagram
  • Each Voronoi vertex corresponds to a Delaunay triangle
  • Each Voronoi edge corresponds to a Delaunay edge
• Properties of Delaunay triangulations:
  • Maximizes the minimum angle among all possible triangulations of the point set
  • Satisfies the empty circle property: no point in the set is inside the circumcircle of any triangle in the triangulation
  • The union of all triangles in the Delaunay triangulation forms the convex hull of the point set
• Delaunay triangulations can be constructed directly using various algorithms
  • Incremental insertion, divide-and-conquer, flip-based algorithms
• The Delaunay triangulation of a set of points in the plane has a total complexity of $O(n)$, where $n$ is the number of points

Algorithms and Implementations

• Fortune's algorithm for constructing Voronoi diagrams:
  • Sweep line algorithm that maintains a beach line and a priority queue of events
  • Time complexity: $O(n \log n)$ for $n$ points
  • Implemented using a balanced binary search tree (e.g., AVL tree or red-black tree) for the beach line and a priority queue for events
• Incremental insertion for constructing Delaunay triangulations:
  • Add points one by one, updating the triangulation at each step
  • Time complexity: $O(n^2)$ for $n$ points, but can be improved to $O(n \log n)$ using a point location data structure
  • Implemented using a data structure for point location (e.g., trapezoidal map or quadtree) and a stack for maintaining the triangulation
• Divide-and-conquer for constructing Delaunay triangulations:
  • Recursively partition the point set, construct triangulations for subsets, and merge them
  • Time complexity: $O(n \log n)$ for $n$ points
  • Implemented using a recursive function and a merge step that updates the triangulation along the merge line
• Flip-based algorithms for constructing Delaunay triangulations:
  • Start with an arbitrary triangulation and flip edges until the Delaunay property is satisfied
  • Time complexity: $O(n^2)$ for $n$ points, but can be improved to $O(n \log n)$ using a randomized incremental approach
  • Implemented using a data structure for the triangulation (e.g., half-edge or quad-edge) and a function for flipping edges

Real-World Applications

• Nearest neighbor search and spatial interpolation
  • Voronoi diagrams can be used to efficiently find the nearest point in a set to a given query point
  • Spatial interpolation methods, such as natural neighbor interpolation, use Voronoi diagrams to estimate values at unsampled locations
• Facility location and resource allocation
  • Voronoi diagrams can help determine optimal locations for facilities (e.g., hospitals, fire stations) to minimize the maximum distance to any point in a region
  • Resource allocation problems, such as assigning students to schools or customers to service centers, can be solved using Voronoi diagrams
• Mesh generation and finite element analysis
  • Delaunay triangulations are often used as the basis for generating high-quality meshes for finite element analysis
  • The empty circle property ensures that the triangles are well-shaped and suitable for numerical simulations
• Path planning and robot navigation
  • Voronoi diagrams can be used to find safe paths for robots or autonomous vehicles, maximizing the distance from obstacles
  • The Voronoi diagram of a set of obstacles defines a roadmap for navigation, with Voronoi edges representing safe paths
• Computational biology and protein structure analysis
  • Voronoi diagrams and Delaunay triangulations are used to study the packing and structure of atoms in proteins
  • The alpha shape, a subset of the Delaunay triangulation, can be used to define the surface and volume of a protein

Advanced Topics and Extensions

• Higher-dimensional Voronoi diagrams and Delaunay triangulations
  • Voronoi diagrams and Delaunay triangulations can be defined in higher dimensions (3D, 4D, etc.)
  • The complexity and algorithms for constructing these structures become more involved as the dimension increases
• Weighted Voronoi diagrams and power diagrams
  • Each point is assigned a weight, and the distance to a point is defined as the power distance (Euclidean distance squared minus the weight)
  • Power diagrams have applications in modeling crystal growth and molecular dynamics
• Constrained Delaunay triangulations
  • Additional constraints, such as prescribed edges or polygons, are imposed on the triangulation
  • Constrained Delaunay triangulations have applications in terrain modeling and mesh generation for complex domains
• Voronoi diagrams and Delaunay triangulations on surfaces
  • Voronoi diagrams and Delaunay triangulations can be defined on surfaces, such as spheres or arbitrary manifolds
  • These structures have applications in computer graphics, such as texture mapping and remeshing
• Kinetic Voronoi diagrams and Delaunay triangulations
  • The points are moving over time, and the goal is to maintain the Voronoi diagram or Delaunay triangulation as the points move
  • Kinetic data structures and event-driven algorithms are used to efficiently update the structures