Discrete Fréchet Distance - Intuitive definition vs. Formal defintion

Question

I'm struggling at the moment, with the definition of, the discrete Fréchet Distance.

I'm thinking about how, the couple pairs are selected:

And I've got this paper: Paper

I'm trying to understand how this paper, could explain me this picture.

Answer 1

The master and his dog walk forward at the parametrised positions $\alpha(t), \beta(t), t\in[0,1]$. At $t=0$ they start and $t=1$ they reach the destination. $f(\alpha(t))$ is the position of the master and $g(\beta(t))$ is the position of the dog (at time $t$). The question is how to minimize the maximum value of $<f(\alpha(t)),g(\beta(t))>$ for some distance $<,>$. In other words we need to find which positions of the master and dog $\alpha\&\beta$ at each point in time which minimizes the maximum distance at any point.

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In a maybe more easy to understand way: A train and a wagon are on different tracks. The train is supposed to pull the wagon with a (strong enough) chain. Which is the minimum length of the chain for the train to not get stuck and not be able to move forward with the wagon?

Or the dog and his master walking on different tracks, what is the minimum length of the leash for the master and dog not get stuck and not arrive at the goal?

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Attempt to explain the figure ( although I'm not sure my interpretation is correct ):

Each couple is the distance at a state $\alpha(t_0),\beta(t_0)$ : $<\alpha(t_0),\beta(t_0)>$. So if we start looking from the left:

1. First they start at the initial state both at their leftmost point, then blue takes a step forward and the new shortest distance is the new line.
2. Then there are no more multiple couplings to the first red so
   1. red takes a step, until there are no couplings from the new red dot so
   2. blue takes a step.
3. Then red takes one step for each of the couplings to the new blue position we are standing at (7 steps in a row).

4. When there are no more couplings we switch who moves (as in step 2)

5. Then next point the graph screws up by accidentally putting a red dot right on the blue line so it's kind of difficult saying what happens then.