I would do the following:
1) build interpolated (e.g. bezier interpolation) curves;
2) subsample interpolated curves: from this we would have data1 = [(x1, y1), ..., (xN, yN)] and data2 = [(p1, q1), ..., (pN, qN)];
3) Compute procrustes distance between data1 and data2. Procrustes distance is a distance between two shapes only (it doesn't account curve sizes and relative rotations).
1) build interpolated (e.g. bezier interpolation) curves;
2) subsample interpolated curves: from this we would have data1 = [(x1, y1), ..., (xN, yN)] and data2 = [(p1, q1), ..., (pN, qN)];
3) Compute procrustes distance between data1 and data2. Procrustes distance is a distance between two shapes only (it doesn't account curve sizes and relative rotations).