How can I read and understand the formula of Hausdorff metric? I understand the concept, but I am not sure what this formula really wants to tell me? I have seen an explanation from this link that,
Link: http://cgm.cs.mcgill.ca/~godfried/teaching/cg-projects/98/normand/main.html
for instance, there are 2 sets of points (color red and blue). For the first red point, it calculates the "shortest" distance from the red point to the nearest blue point then saves that 1st data. For the next red point, it then calculates the "shortest" distance from that red point to the nearest blue point and saves that 2nd data. Then the 1st and 2nd data will compete to each other and returns the "largest" distance.
Another question is that, why do we have to do this if we can just use the basic Euclidean distance? What is the sole purpose?
$$ H = \max\left\{ \max_{x \in A} \min_{y \in B}\|x-y\| , \max_{x \in B} \; \min_{y \in A}\|x-y\| \right\} $$
$H$ is the larger of the greatest distance to $B$ from points of $A$ and the greatest distance to $A$ from points of $B$. If there is an open ball of radius $r$ centred at a point of one of these sets and disjoint from the other set, then $H \ge r$.