Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways
Fig 1
Comparison of Hausdorff and Fréchet distance. Two paths P (green) and Q (cyan) begin at state c0 and end at state cf with directionality indicated by the arrows.
The Fréchet distance δF and Hausdorff distance δH are given by the lengths of the purple and orange lines, respectively. The purple lines are the same length and correspond to the minimally stretched Fréchet “leash”; the orange line spans a pair of points separated by the Hausdorff distance (only one is shown because in this case there are infinitely many pairs of points with the same δH). Due to the backtracking of path P toward state A, combined with the monotonicity (no-backward-movement) constraint of the Fréchet metric, δF > δH.