How to represent a polygon with hole(s)?

Question

It\'s usually popular to work with polygons with their vertices sorted CW or CCW in vectors(2*1 or 1*2 matrices). However, how to state polygons with holes in vectors?

Answer 1

As others have mentioned, a polygon with holes can be represented as an exterior boundary, plus zero or more interior boundaries, all of which are mutually nonoverlapping*. If you use nonzero winding number to determine inside/outside, be sure to specify your interior boundaries in the opposite direction as the exterior boundaries (counterclockwise for exterior and clockwise for interior, or vice-versa) so that the contour integrals are zero inside the holes.

FYI, tis kind of definition/representation has been formalized in the OpenGIS Simple Features Specification (PDF).

As far as representation:

I'd probably have a cell array of K Nx2 matrices, where the first element in the cell array is the exterior boundary, and the remaining elements (if any) in the cell array are the interior boundaries. I would use a cell array because there may not be the same number of points on each boundary.

*nonoverlapping = except at individual points, e.g. a diamond inside a square: