Index express the similarity in shape between two data sets

Question

I have two data sets. One is points in the blue curve. The other is points in the green curve. Now I want to find a property (P) or something else like index that has the following characteristics:

1. It represents the similarity between two data sets. (With similarity I am referring to the same shapes plot by these points of two data sets)
2. It doesn't depend on the scale. Two shapes have the same shape but different scale still give the same P value.

Could you suggest what P should I consider?

Answer 1

I've tried to use affine transformation to match shapes and used, area of symmetric difference between target and match shapes (D) to calculate

P = 1 - D/AreaOfShapeToMatch:

Result looks like this:

To find transformation coefficients using least squares technique shouldn't be a challenge if you know how to match points.

You cannot use this technique to compare triangles, because any 2 will match. The same with rectangular shapes. Rectangle will match square.