Simple Polygon Orientation

From CGAFaq

Compute the signed area of the polygon. The orientation is counter-clockwise if this area is positive.

A slightly faster method is based on the observation that it isn't necessary to compute the area. Find the lowest vertex (or, if there is more than one vertex with the same lowest coordinate, the rightmost of those vertices) and then take the cross product of the edges fore and aft of it. Both methods are O(n) for n vertices, but it does seem a waste to add up the total area when a single cross product (of just the right edges) suffices. Code for this is available at ftp://cs.smith.edu/pub/code/polyorient.C (2K).

The reason that the lowest, rightmost (or any other such extreme) point works is that the internal angle at this vertex is necessarily convex, strictly less than pi (even if there are several equally-lowest points).