Besides being a topic of interest of its own, the generation of random polygons has two main areas of application: a) testing the correctness and b) evaluating the CPU-time consumption of algorithms that operate on polygons. For testing the correctness of an algorithm, the goal is to generate a diverse set of input data such that all branches of the algorithm will be executed with a high probability. The same motivation applies to the practical testing of an algorithm's CPU-consumption. Ideally, one would like to test the performance of an algorithm on data of practical relevance. However, it often is next to impossible to obtain a sufficiently large number of practically relevant inputs. Then the second-best choice is to run an algorithm for a reasonably large number of random inputs.

This need for polygonal test data motivated
Thomas Auer
and
me
to study the generation of random polygons on a given set of vertices.
Since no polynomial-time solution is known for the uniformly random
generation of simple polygons, we have focused on heuristics
that offer a good time complexity and still generate a rich variety of
different polygons. Our algorithms were implemented by
Thomas Auer
and became part of **RPG**, our Random Polygon Generator.
RPG currently features heuristics (and algorithms) for the generation of
x-monotone, star-shaped, and general simple polygons. In addition, polygonal
approximations of "random" free-form curves can be generated via a smoothing
option.

More recently, Martin Heimlich ported RPG's GUI to GTK and added a neat heuristic for generating a "random" multiply-connected polygonal area for a given set of vertices. Roughly, we start with generating a random polygon on the set of vertices and then cut-off some of the pockets of the polygon by inserting parallel bridge edges, thus generating polygonal holes within a polygonal outer boundary. Furthermore, Michael Gschwandtner added code for geometric hashing in order to speed up the search for polygon edges that intersect pairwise. This new code provided a substantial improvement of the practical efficiency of those heuristics for the generation of random polygons that first generate an arbitrary polygon and then look for intersections, such as "2-Opt Moves".

**RPG** is available upon request.
Please
contact me,
and let me know that you would like to get it.

T. Auer, M. Held (1996):

``Heuristics for the Generation of Random Polygons''.

Proc. 8th Canad. Conf. Computat. Geometry, Carleton University Press, F. Fiala, E. Kranakis, J.-R. Sack (eds.), pp. 38-44; Ottawa, Canada, Aug 12-15, 1996.

The following images show random polygons which were generated by means of RPG. (Click on an image icon in order to see the full-size image. The full-size images have 800x900 pixels.)

This image shows a random star-shaped polygon with 50 vertices. |

This image shows a "random" simple polygon with the same set of 50 vertices. The polygon was computed using RPG's "2-Opt Moves" heuristic. |

This image shows a "random" multiply-connected polygon with the same set of 50 vertices. (Holes are shown in cyan.) |

This image shows the result of "smoothing" the previous simply-connected polygon three times. (This is a fairly smooth polygon on 400 vertices.) |

This image shows the result of "smoothing" the previous multiply-connected polygon two times. |

This image shows a "random" simple polygon on a set of 200 clustered vertices. The polygon was computed using RPG's "2-Opt Moves" heuristic. |

This image shows the result of "smoothing" the previous polygon four times. |

This image shows a "random" simple polygon on a set of 10000 clustered vertices. The polygon was computed using RPG's "2-Opt Moves" heuristic. |

This image shows a zoomed view of the previous polygon. |

This image shows a zoomed view of a multiply-connected polygon on the same set of 10000 clustered vertices. (Holes are shown in cyan.) |

file last modified: Friday, 26-Feb-2016 12:34:25 CET
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