# NAME

Math::Geometry::Voronoi - compute Voronoi diagrams from sets of points

# SYNOPSIS

```
use Math::Geometry::Voronoi;
# load a set of points
my @points = ([1, 2],
[1, 3],
[2, 2],
[0, 1],
[0, 10],
[0.5, 11]);
my $geo = Math::Geometry::Voronoi->new(points => \@points);
# compute your diagram
$geo->compute;
# extract features
my $lines = $geo->lines;
my $edges = $geo->edges;
my $vertices = $geo->vertices;
# build polygons
my @polygons = $geo->polygons;
```

# DESCRIPTION

This module computes Voronoi diagrams from a set of input points. Info on Voronoi diagrams can be found here:

` http://en.wikipedia.org/wiki/Voronoi_diagram`

This module is a wrapper around a C implementation found here:

` http://www.derekbradley.ca/voronoi.html`

Which is itself a modification of code by Steve Fortune, the inventor of the algorithm used (Fortune's algorithm):

` http://cm.bell-labs.com/who/sjf/`

I made changes to the C code to allow reading input and writing output to/from Perl data-structures. I also modified the memory allocation code to use Perl's memory allocator. Finally, I changed all floats to doubles to provide better precision and to match Perl's NVs.

# INTERFACE

## new

```
my @points = ([1, 2],
[1, 3],
[2, 2],
[0, 1],
[0, 10],
[0.5, 11]);
my $geo = Math::Geometry::Voronoi->new(points => \@points);
```

Create a new object, passing in a single required parameter called 'points'. This must be an array or arrays containing at least two values each, the X,Y values for your points. Any extra data will be ignored.

## points

Returns the *sorted* set of points used by the voronoi algorithm. This is the ordering refered to by the lines() output below.

## compute

Call this to build the diagram. Returns nothing.

## lines

Returns an array ref containing arrays of lines in the output diagram. The data by index:

```
0: the a value in the ax + by = c equation for the line
1: the b value
2: the c value
3: the index of one point for which this line is the bisector.
4: the index of the other point for which this line is the bisector.
```

Note that 3 and 4 are not the end-points of the line - they are points perpendicular to the line. Either 3 or 4 may be -1 meaning no point.

## vertices

Returns an array ref containing arrays of vertices in the output diagram. These are the points which connect edges running along the lines. The data by index:

```
0: the x value
1: the y value
```

## edges

Returns an array ref containing arrays of edges in the output diagram. An edge is defined as a segment of a line running between two vertices. The data by index:

```
0: the index of the line
1: the index of vertex 1
2: the index of vertex 2
```

Either 1 or 2 can be -1 meaning "infinite".

## polygons

` @polys = $geo->polygons();`

This method attempts to assemble polygons from non-infinite edges in the diagram. This part of the code is written in Perl and is of my own invention. I needed this facility in order to color the diagrams created by this module. It seems to work reasonably well for my uses but I'm sure it's nowhere near the quality of Steve Fortune's code! Feedback welcome.

This method returns a reference to an array containing first a point index and then a list of vertex coordinates. The point is the point inside the polygon and the vertices are in drawing order for the closed polygon surrounding the point. For example:

` @polys = ( $point_index, [$lat1, $lon1], [$lat2, $lon2], ... );`

One optional parameter is available - normalize_vertices. This option is necessary because the algorithm used needs to match up points from one edge to another and doing that with floating point numbers requires some kind of normalization (otherwise 1.1 != 1.10001). For example, if your coordinates are on an integer grid you might do:

` @polys = $geo->polygons(normalize_vertices => sub { int($_[0]) });`

Or if you're using floating point and your coordinates are useful down to 2 decimal places:

` @polys = $geo->polygons(normalize_vertices => sub { sprintf("%.2f", $_[0]) });`

The point is to produce coordinates in a format where they will compare as equal textually, side-stepping the problem of comparing floats numerically.

# TODO

Possible projects, if you're in the mood to help out:

```
- Add the ability to combine polygons based on a mapping of
same-type points. Map overlays get cluttered by internal lines
with you're coloring multiple polygons the same. All edges
connect exactly two polygons, so this should be relatively easy.
Sadly, my limited math skills have thwarted me on this one - I
spent several days but ultimately couldn't get it working reliably
on all possible shapes.
- Remove the need for the normalize_vertices option to polygons(),
somehow (fuzzy matching?).
- Setup a site where people can play with the module visually and
see purty colors. Could be an excuse to play with the new canvas
stuff in modern browsers.
- Add tests that actually examine the output for sanity. So far the
tests just look at the format and range of the output data - to
see if it's actually doing a decent diagram I look at graphical
output.
```

# AUTHOR

Sam Tregar <sam@tregar.com>

# COPYRIGHT AND LICENSE

As far as I can tell the underlying C code used here never had a license attached to it, or if it did I couldn't find any trace of it. If this worries you please contact Steve and Derek through the links above.

The Perl and XS code in this library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.8.5 or, at your option, any later version of Perl 5 you may have available.