Create largest circle (maximum radius) inside a polygon in ArcGIS for Desktop?

Question

I am trying to create the largest possible circle inside a polygon including its perimeter and area size. I created via minimum boundary geometry the smallest possible circle outside a polygon (see image) and want to do some calculation with its largest circle inside the polygon. I have a shapefile of 25 polygons. Each has to have its own largest circle. This circle can be anywhere in the polygon, not necesserily at the centre since my polygons are uneven.

I tried looking at Find maximum radius of circle that will fit within an irregular polygon? but it was no success. I have the same problem as the OP.

Answer 1

I have worked up an answer in ArcMap and using ET Geowizards (as it is what I have access to but I think the same tools exist in ArcMap but require more than a basic license):

• Convert polygon to point using 'Deepest Point' option (i.e. maximising distance from the edge of the polygon giving you the most room for the circle.
• Buffer points using the 'ET_Depth' attribute generated by the previous tool (Initially I spent some time trying to use some sort of 'nearest' tool to calculate the shortest distance between the point and he edge of the polygon but it turns out this is given by the first tool, you may need to do this if you don't want to use ET geowizards).

I haven't fully tested this so, maybe I missed a polygon shape that wouldn't work (conceptually I can't think of one). The bottom left one in the rectangle is interesting because the circle could be anywhere down the polygon and be just as valid. I guess it is at the top because the tool start works systematically top left to bottom right.

Answer 2

Some term this is as girth and the University of Connecticut, Center for Land Use Education & Research, has developed a free geoprocessing tool (Shape Metrics) to calculate this among many other shape metrics. This is mentioned in the link as an index but it is quite easy to find the radius from the formulae given. You can download it from the link, https://clear.uconn.edu/tools/Shape_Metrics/Shape%20Metrics.zip

Two important points (after Hornbydd's comment), the radius derived may not be the exact solution but a good-enough approximation and the tool does not give you the centre of the girth (centroid of circle) or create a circle polygon as output.

Answer 3

This is basically a skeletonization problem.

Calculate the shortest distance from every interior point to the edge of the polygon. The maximum of this set of values is the radius of the largest circle that will fit in the polygon and any interior point which has this maximum value is a valid center for that circle. There may be more than one center and more than one valid circle. In fact, for any polygon except a perfect circle, there may be one or more connected lines (the "skeleton") along which an infinite number of points lie that are at this maximum distance. Since there are infinite interior points, a grid or regularly-spaced point array of the the appropriate precision, say 0.5 feet or 0.5 meters, will yield a precise approximation.

The pure vector approach would be to create a regular rectangular or triangular array of points and then calculate the distances from all of those points interior to the polygon to its edge. Find the points with the largest distances (rounded to the same precision as the mesh). Use these points to draw your circles (a buffer of the distance).

You could also use a hybrid raster/vector approach. First, convert the polygon and its edge to raster grids, Second, calculate the minimum euclidean distance from each polygon cell to the edge raster cells. Third, get the maximum of the minimum distances. Fourth, convert the maximum cells back to points and use them to draw your circles.

[EDITED]

The approach described above is essentially the one used by the Girth metric from ShapeMetrics mentioned in an other answer. From the the girth for each polygon a negative buffer of the polygon to the girth minus the precision desired will yield the polygon eroded by the girth and the contained centroid of the eroded polygon is an ideal location for the circle.