How to determine if a list of polygon points are in clockwise order?

Some of the suggested methods will fail in the case of a non-convex polygon, such as a crescent. Here's a simple one that will work with non-convex polygons (it'll even work with a self-intersecting polygon like a figure-eight, telling you whether it's mostly clockwise).

Sum over the edges, (x₂ − x₁)(y₂ + y₁). If the result is positive the curve is clockwise, if it's negative the curve is counter-clockwise. (The result is twice the enclosed area, with a +/- convention.)

point[0] = (5,0)   edge[0]: (6-5)(4+0) =   4
point[1] = (6,4)   edge[1]: (4-6)(5+4) = -18
point[2] = (4,5)   edge[2]: (1-4)(5+5) = -30
point[3] = (1,5)   edge[3]: (1-1)(0+5) =   0
point[4] = (1,0)   edge[4]: (5-1)(0+0) =   0
                                         ---
                                         -44  counter-clockwise

Find the vertex with smallest y (and largest x if there are ties). Let the vertex be A and the previous vertex in the list be B and the next vertex in the list be C. Now compute the sign of the cross product of AB and AC.

References:

• How do I find the orientation of a simple polygon? in Frequently Asked Questions: comp.graphics.algorithms.

• Curve orientation at Wikipedia.