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Problem Description

The question is: "What is the largest rectangle that can be inscribed in an arbitrary given simple polygon"?

Background & Techniques

This is an exploration of techniques for finding the maximum area rectangle that can be inscribed in arbitrary concave or convex simple polygons.  Polygons can be defined by clicking vertex points on the image area or loaded from a text file.  All coordinates in this version are pixel locations on an  500 x 500 pixel image area.  

Once a polygon has been defined or loaded, the search strategy is based on finding the maximum area polygon from a given center point.  A trial and error loop successively creates rectangles with a range of major axis angles (typically 0 to 180 degrees)  and a range of Width/Height ratios (typically 1 to 5).   Parameters can be set controlling angle and ratio step size and the max ratio to be tested. 

Several strategies are available for searching for rectangles.  The "Strategies" options create trial rectangle centers which are then passed to the rectangle creation procedure, looking for new maximum area examples.

Load and Save buttons do what they say, preserving and restoring polygon vertices and maximal rectangle information if a search has been performed.  There are additional options to ignore best rectangle for a particular polygon to aid in evaluating specific strategies, and one to restrict rectangles to squares (Width/Height ratio set to 1.0) .

Note: This is not a finished project - there are several known rough spots to be polished as time permits.
 

Programmer's Notes:

A small book could be written describing the code written so far, however because of the exploratory nature of the program, the geometry involved, and time constraints it is probably best to delay better documentation  until the code has evolved to a more permanent form.  Much of the code was developed while "in the zone" which is not conducive to tidy organization so cleanup is needed. 

 It is still fun to play with though, watching various strategies improve results, sometimes quickly, sometimes slowly.  

The UGeometry library unit has been modified to add a TPolygon type and to override the PolygonArea function to return the Centroid (center of mass) point as well as the Area of a polygon.   The modified version has been included with the source download here and will be added to our DFFLib zip file with the next update.        

Running/Exploring the Program 

Suggestions for Further Explorations

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