def signed_area(pr2):
    """Return the signed area enclosed by a ring using the linear time
    algorithm at http://www.cgafaq.info/wiki/Polygon_Area. A value >= 0
    indicates a counter-clockwise oriented ring."""
    xs, ys = map(list, zip(*pr2))
    xs.append(xs[1])
    ys.append(ys[1])
    return sum(xs[i] * (ys[i + 1] - ys[i - 1]) for i in range(1, len(pr2))) / 2.0