Calculating roundness/compactness of polygon?

Question

I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.

The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.

So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.

But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.

For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?)

Answer 1

The compactness of an object can be measured using the Polsby-Popper test by determining the Polsby-Popper (PP) score.

The PP score is determined by: multiplying the polygon's area by 4pi and dividing by the perimeter squared. Using this, a circle will have a score of 1 and any other geometric shape has a smaller ratio.

disc :(4*PI)* PI*R² / 4PI²R²= 1

square : (4*PI) * C² / 16 * C² = PI/4 ~=0.78

Another usefull index could be the lenght/width of the smallest enclosing rectangle (see the minimum bounding geometry tool). But in this case the square and the circle are alike and concavity is ignored.

As a last recommendation, if you work with perimeter, it is usefull to "smooth" your object before computing the indices, in order to avoid "fractal" effect (especially if your polygons come from raster to polygon conversion)

Answer 2

In addition to the roundness formula outlined by radouxju in his response and others like the ratio of the area of the polygon to the area of its minimum bounding circle - ST_Area(geom)/(ST_Area(ST_MinimumBoundingCircle(geom)) as rnd_check in PostGIS: I find it often helps to check the number of vertices/points in a 'suspicious' geometry - ST_NPoints(geom) in PostGIS.

The stuff I do is different from what you're describing, but I find that the NPoints filter helps discriminate between property parcels (which can actually be long and skinny, of course) and rivers and other long-skinny natural features. There's the odd long, skinny property parcel that borders a river, but anomaly-checking is why we get paid the big bucks (HA!): they invariably have one side (at least) that's straight for a goodly portion of the feature's length, so job's done.

Also, it's rare (in my workflow) that there's not some aspatial identifier that can't be brought to bear, and in any case a lot of my work is set up so that we're analysing 'delta' (changes across time) so if the data at T=0 is clean and nobody's introduced 'negative enhancements' for T ∈ [0, t-1], then delta for an entire state for T=t|t-1 can be done in half a day.

Loads of points in a thing that's long and skinny and doesn't have any single line that's ... probably a river.

A reservoir with a dendritic shape would certainly challenge that guess, but it's likely that filtering on the overall length of the shape might yield results if the entire river is one polygon (we should be so lucky) - or find the number of paths from the 'narrow' end to the fat end (the number of branches).

Lake vs park... I would try to do that by aerial/satellite imagery, given my druthers: it's a way easier problem to use a land/water classifier where the region to be checked is known, than trying to identify and extract a water region from an image where the location of the water is not known.

I also found this answer (to a different question) very useful for discriminating between long-skinny features.

Answer 3

I was facing a similar problem, and ended up solving the equation for both circumference and area of a circle to make them equal one another like so:

2*pi*r = c <=> r = c/2*pi pi*r^2 = a <=> r = sqrt(a/pi)

c/2*pi = sqrt(a/pi) <=>

sqrt(a/pi)

-------------- = 1

c/2*pi

This index is between 0 and 1 where 1 is a perfect circle. I don't know is this is an established method but I would love to hear from anyone who might have seen it elsewhere.

Answer 4

WhiteBox Tools has a suite of tools for computing shape metrics have a look at the manual here. If you are an ArcGIS user they expose the tools in a user friendly toolbox, this can be downloaded here.