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Posted 10 Aug 2005

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! 


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.


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).


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


About the Author

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.

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Comments and Discussions

GeneralRe: My vote of 5 Pin
Member 868370514-May-12 22:38
memberMember 868370514-May-12 22:38 
GeneralRe: My vote of 5 Pin
maurice.calvert15-May-12 5:56
membermaurice.calvert15-May-12 5:56 
Generalplease help soon Pin
daskan8-Aug-09 2:55
memberdaskan8-Aug-09 2:55 
QuestionVoronoi Cells Pin
C-Bl28-Jul-09 0:08
memberC-Bl28-Jul-09 0:08 
Answer3 data points case Pin
SOAD_21-Jun-09 7:08
memberSOAD_21-Jun-09 7:08 
GeneralRe: 3 data points case Pin
rhill-ca26-Jun-09 7:45
memberrhill-ca26-Jun-09 7:45 
GeneralThiessen Polygons Pin
SOAD_19-Jun-09 15:46
memberSOAD_19-Jun-09 15:46 
GeneralAnother wrong vertex Pin
Sunil Terkar11-May-09 3:04
memberSunil Terkar11-May-09 3:04 

For following another set of points, algorithum misses the vertex by almost 250 mt.

6 10494.00016 10092.53629
7 10290.7531 10684.97519
8 9947.773702 11043.95948
9 9776.284008 11059.26798
10 10991.95541 10217.30055
11 10057.01902 10092.5363

It gives 8 edges and 3 vertex, but due to wrong location of vertex 3, egde no 6,7 and 8 have not come out properly. Actual location of vertex 3 should be 9819.40,10578.40. I strongly feel that it should have 4 vertexes.

Could you please look into it.
Sunil Terkar
GeneralRe: Another wrong vertex Pin
BenDi11-May-09 11:03
memberBenDi11-May-09 11:03 
GeneralRe: Another wrong vertex Pin
Sunil Terkar12-May-09 1:37
memberSunil Terkar12-May-09 1:37 
GeneralRe: Another wrong vertex Pin
Sunil Terkar12-May-09 6:09
memberSunil Terkar12-May-09 6:09 
GeneralRe: Another wrong vertex Pin
rhill-ca26-Jun-09 7:34
memberrhill-ca26-Jun-09 7:34 
GeneralShort of one edge and one vertex Pin
Sunil Terkar6-May-09 0:11
memberSunil Terkar6-May-09 0:11 
GeneralRe: Short of one edge and one vertex Pin
rhill-ca26-Jun-09 7:31
memberrhill-ca26-Jun-09 7:31 
Generalmedial axis Pin
led16-Apr-09 10:38
memberled16-Apr-09 10:38 
GeneralThanks! Pin
Reconstructivism30-Mar-09 3:48
memberReconstructivism30-Mar-09 3:48 
GeneralRays Pin
Wayne Romer22-Mar-09 21:55
memberWayne Romer22-Mar-09 21:55 
GeneralRe: Rays Pin
Reconstructivism30-Mar-09 3:56
memberReconstructivism30-Mar-09 3:56 
GeneralRe: Rays Pin
BenDi30-Mar-09 9:57
memberBenDi30-Mar-09 9:57 
GeneralRe: Rays Pin
Pavel Vladov29-Apr-09 1:54
memberPavel Vladov29-Apr-09 1:54 
GeneralRe: Rays Pin
BenDi29-Apr-09 10:49
memberBenDi29-Apr-09 10:49 
GeneralRe: Rays Pin
Pavel Vladov30-Apr-09 0:39
memberPavel Vladov30-Apr-09 0:39 
GeneralRe: Rays Pin
BenDi30-Apr-09 9:04
memberBenDi30-Apr-09 9:04 
GeneralRe: Rays Pin
Pavel Vladov30-Apr-09 23:36
memberPavel Vladov30-Apr-09 23:36 
GeneralRe: Rays Pin
Slagh14-May-09 23:10
memberSlagh14-May-09 23:10 

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