FortuneSwift

FortuneSwift is a swift framework that uses Fortune's Algorithm to generate Voronoi diagrams and Delaunay triangulations from a set of points on a 2D plane. Fortune's Algorithm has a time complexity of O(n log n), and a space complexity of O(n). This framework is compatible with iOS 8+, macOS 10.13+.

Examples

Installation

The FortuneSwift framework can be installed using the Swift Package Manager. In Xcode: File > Swift Packages > Add Package Dependency.. with URL: https://github.com/TateLiang/FortuneSwift.git

Or alternatively:

• Add .package(url: "https://github.com/TateLiang/FortuneSwift.git", from: "1.1.6") to your Package.swift file's dependencies.
• Update your packages using $ swift package update.

Usage

Setup

1. Add package dependancy
2. Import import FortuneSwift
3. Create a Voronoi object, specifying an array of Coordinates and the rectangular area.

let points: [Coordinate] = [ ... ]
let rect = (x: 25, y: 25, width: 500, height: 500)

let voronoi = Voronoi(sites: points, numPoints: points.count, rect: rect)

Alternatively, if the array is not given, numPoints of random points are generated within the rectangular area.

let voronoi = Voronoi(numPoints: 150, rect: rect)

Processing Graph

FortuneSwift outputs a doubly connected edge list (DCEL) stored as arrays of sites, voronoi vertices and half-edges. This framework provides four classes for accessing the diagram: Coordinate, Site, Vertex and HalfEdge. This information can be used to draw a Voronoi diagram or Delaunay triangulation.

Once the voronoi object has been created, the resulting arrays can be obtained via the properties:

var voronoiVertices: [Vertex]
var voronoiEdges: [HalfEdge]
var voronoiSites: [Site]

The main properties of each class are detailed below.

Site

var x: Double
var y: Double
var firstEdge: HalfEdge?
var surroundingEdges: [HalfEdge]?

Vertex

var x: Double
var y: Double
var incidentEdges: [HalfEdge]

HalfEdge

var origin: Vertex?
var destination: Vertex?

var twin: HalfEdge?
var next: HalfEdge?
var prev: HalfEdge?

var incidentSite: Site?

func walk() -> [HalfEdge]? 

Notes

• firstEdge is an arbitrary edge defining the voronoi cell of a site. surroundingEdges can be used for a list of all edges defining the cell.
• The HalfEdges trace out a voronoi cell counter-clockwise with the incidentSite to its left, assuming a coordinate system where y increases upward and x increases rightward.
• incidentSite is nil if it is an edge on the exterior border of the bounding rectangle.
• walk() outputs an ring of the HalfEdges defining the cell, or nil if the edges don't form a ring (should not happen)

Details

Problem Definition

A voronoi diagram is a tesselation of cells in a plane, representing the points closest to a particular point. For each voronoi "site", its region is defined as the points closer to it than any other site. Voronoi diagrams are also dual-graphs of the Delaunay triangulation, where each edge of the Voronoi diagram corresponds to an adjacent edge in the Delaunay triangulation between the two incident points. Voronoi diagrams have a variety of applications including in Astronomy, Art, Biology, Robotics, and Physics.

Algorithm

Fortune's Algorithm is a sweep line algorithm in computational geometry. Like problems such as convex hulls or segment intersection, an event queue is maintained through a priority queue data structure. Fortune's Algorithm makes use of a binary search tree to also maintain a "beach line", representing the currently known cells based on the location of the sweep line. At the end of the algorithm, the infinite edges can be bounded by a polygon, in this case, a rectangle.

Issues

• Points outside of the bounding box will still effect the voronoi diagram, as the box simply bounds all of the infinite edges.
• The algorithm does not work with the edge case where all points are collinear.
• The algorithm does not work with the edge case where there are multiple sites at the same location.