How to calculate the area of multiple overlapping right-angled polygons?

A classical way of computing such an area is to use a sweep algorithm. You can have a look at question Area of overlapping rectangles to get a description of the algorithm in the simpler case of rectangles Then you can either decompose your polygons into rectangles, or adapt the sweep algorithm so that this decomposition is done implicitly during the sweep.

A classical way of computing such an area is to use a sweep algorithm. You can have a look at question Area of overlapping rectangles to get a description of the algorithm in the simpler case of rectangles. Then you can either decompose your polygons into rectangles, or adapt the sweep algorithm so that this decomposition is done implicitly during the sweep.

Another way to do it would be to calculate area using a contour integral. Walk around the perimeter of the area and calculate area using Green's theorem and numerical quadrature. Easy peasy.

The problem with this approach is that finding the perimeter is not an easier task than finding the area. – Camille Sep 8 '09 at 23:26 Interior nodes are shared by four polygons if they're all quadrilaterals. A perimeter node is shared by one, two, or three polygons.

You can identify the perimeter nodes and start your walk at any point by transitioning to the next perimeter node. Integrate along that edge and then find the next one. – duffymo Sep 8 '09 at 23:32 I do not see why interior nodes should be shared by four polygons.

Take for example two overlapping rectangles (each of them covering one corner of the other one). You also have to identify nodes of the perimeter that are not nodes of the original polygons. Of course you can identify the boundary of the union, but you will have to resort to some kind of sweep algorithm anyway for that task, hence my comment.

– Camille Sep 9 '09 at 7:09.

If you can represent the polygons as 2D int or bool arrays (Aij == 1 if the ijth slot is contained in the polygon, 0 otherwise) then you can create the union of the polygons on a larger 2D "bounding" array. The algorithm is something like this: // get your polygons, each represented by a 2D array as described above // create a "bounding" array B that can contain all polygons (watch out for // sparsely spaced polygons in which case B could be large) // populate B s.t. Bij == 1 if ijth slot is contained in // the union of all polygons // count all slots in B where Bij == 1 (this will be the "common" area) Thinking about the run time and space requirements: need to traverse every slot of every polygon.

Need to traverse every slot of the bounding array B. Worst case for space is when the intersection of all polygons is empty (and perhaps they are sparsely spaced). Might be worthwhile to check the empty intersection case first... then if the intersection is empty the "common" area is just the sum of the individual areas.

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