I'm really struggling to find a good Similarity/Distance metric between two vector shapes. E.g., something that would indicate mutual information or edit distance—such that I could quantify the distance between a particular shape instance and a prototypical shape.

In my case, I am trying to quantify the difference in shape between outlines of rock carvings on Stonehenge (purported to be axeheads) and outlines of several different styles of actual bronze age axeheads.