How do I calculate the area of a 2d polygon?

Question

Assuming a series of points in 2d space that do not self-intersect, what is an efficient method of determining the area of the resulting polygon?

As a side note, this is not homework and I am not looking for code. I am looking for a description I can use to implement my own method. I have my ideas about pulling a sequence of triangles from the list of points, but I know there are a bunch of edge cases regarding convex and concave polygons that I probably won't catch.

Answer 1

Here is the standard method, AFAIK. Basically sum the cross products around each vertex. Much simpler than triangulation.

Python code, given a polygon represented as a list of (x,y) vertex coordinates, implicitly wrapping around from the last vertex to the first:

def area(p):
    return 0.5 * abs(sum(x0*y1 - x1*y0
                         for ((x0, y0), (x1, y1)) in segments(p)))

def segments(p):
    return zip(p, p[1:] + [p[0]])

David Lehavi comments: It is worth mentioning why this algorithm works: It is an application of Green's theorem for the functions −y and x; exactly in the way a planimeter works. More specifically:

Formula above =
integral_over_perimeter(-y dx + x dy) =
integral_over_area((-(-dy)/dy+dx/dx) dy dx) =
2 Area