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Fortune's Voronoi algorithm implemented in C#

, 21 Apr 2013
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A C# implementation of the Fortune algorithm to compute 2D Voronoi graphs.

 

NOTE: Code has moved to Google Code! 

Introduction 

Given a set of two dimensional vectors (or data points), a Voronoi graph is a separation of those points into compartments where all points inside one compartment are closer to the contained data point than to any other data point. I won't give any demonstration here, but if you want to know more about Voronoi graphs, check out this.

The applications of Voronoi graphs are quite broad. Very useful for a lot of optimization problems (in most cases, the Delaunay Triangulation which can be easily derived from a Vononoi graph is used there), it ranges to computing topological maps from bitmaps.

[This is an article for freaks. After a rather painful experience writing the thing I hope it will benefit everyone who is looking for this algorithm in a civilized language (or simply does not want to use Fortune's original C implementation).]

In 1987, Steve Fortune described an algorithm to compute such a graph by using a sweep line in combination with a binary tree. A PowerPoint explanation of the algorithm (the one I used to implement it) can be found here. Note that I did not use the linked data structure to represent a graph - I think that is an unnecessary difficulty in the age of ArrayLists and HashSets.

The Implementation

Data points are represented by my own Vector class. It can do much more than needed here (but there was no reason to strip it before bringing it) but I won't explain it here. The most important fact is that although working with doubles the Vector class automatically rounds values to 10 digits (or whatever is set in the Vector.Precision field). Yes, sadly, this is very important if you want to sort of compare doubles.

A VoronoiGraph is a class that only contains a HashSet of vertices (as 2D vectors) and a HashSet of VoronoiEdges - each with references to the left and right data point and (of course) the two vertices that bound the edge. If the edge is (partially or completely) unbounded, the vector Fortune.VVUnknown is used.

BinaryPriorityQueue is used for the sweep line event queue.

Usage

The algorithm itself (Fortune.ComputeVoronoiGraph(IEnumerable)) takes any IEnumerable containing only two dimensional vectors. It will return a VoronoiGraph. The algorithm's complexity is O(n ld(n)) with a factor of about 10 microseconds on my machine (2GHz).

License

This article, along with any associated source code and files, is licensed under The Mozilla Public License 1.1 (MPL 1.1)

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About the Author

BenDi
Software Developer (Senior)
Germany Germany
I did my diploma in Dresden and Sydney where I dealt with algorithms, agents and other cool AI stuff. Now I moved to Frankfurt to work on my PhD dealing with software structures for artificial intelligence systems. If I can, I do things in C# and ASP.NET, but if I have to, my C++, Java and SQL are not that bad.
Long Live .NET.

Comments and Discussions

 
AnswerRe: Getting regions from graph PinmemberBenDi6-Feb-13 23:50 
GeneralRe: Getting regions from graph PinmemberBixel7-Feb-13 12:06 
QuestionSeparating braches of a medial axis sampled from inner voronoi points of a polygon PinmemberMember 917583217-Jul-12 16:19 
QuestionFinding distance for VoronoiGraph vertizes representing Medial Axis of closed boundary PinmemberMember 917583214-Jul-12 5:02 
I am attempting to use this Voronoi algorithm for medial axis computation of a closed boundary of lines, curves, or splines. I have sampled each boundary edge type to create a point at some distance value for a line or arc length of a curve. I then pass these set of points, which represent a filtered closed polygon of the boundary, into the ComputeVoronoiGraph function. I am returned VoronoiGraph vertizes that represent the sampled medial axis.
 
One of the other challenges is that I would like to fit lines, arcs, or splines to the vertizes. But that is a different story...
 
By definition a medial axis is represented as the center points of all inscribed circles that fit within the boundary, which is what I currently have. Although what I would like to also find is the radius of each of the vertizes (inscribed circle center points) that are returned from the ComputeVoronoiGraph function. Any pointers on how to accomplish this? Can I get this information from the algorithm somehow?
 
My intention is to compute the medial axis and filter it similar to the Scale Axis Theorem
GeneralRe: Finding distance for VoronoiGraph vertizes representing Medial Axis of closed boundary PinmemberBenDi15-Jul-12 0:27 
AnswerRe: Finding distance for VoronoiGraph vertizes representing Medial Axis of closed boundary PinmemberKenneth Haugland17-Jul-12 9:31 
GeneralRe: Finding distance for VoronoiGraph vertizes representing Medial Axis of closed boundary PinmemberMember 917583217-Jul-12 16:11 
QuestionIs there a way to construct polygons from all the edges ? And how ? Pinmemberseb.493-Jun-12 21:36 
QuestionPossible issue. When I draw 2 points I haven't 2 area Pinmemberseb.4930-May-12 23:59 
AnswerRe: Possible issue. When I draw 2 points I haven't 2 area PinmemberBenDi31-May-12 3:11 

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