Computation of normalized first derivative in discrete case
For the first part of your question, the significance is just to guarantee that the vector $(\hat{x}'_t,\hat{y}'_t)$ has length&nbsp;$1$. This means that it tells us in which direction $(x_t,y_t)$ is changing, but not by how much. For example, $(\hat{x}'_t,\hat{y}'_t)=(1,0)$ tells us that $x_t$&nbsp;is increasing and $y_t$&nbsp;is constant; $(\hat{x}'_t,\hat{y}'_t) = (-\sqrt{2}/2, \sqrt{2}/2)$ tells us that $x_t$&nbsp;is decreasing, and $y_t$&nbsp;is increasing at the same rate.

The second part isn't something I know about, so I'll leave that for somebody else.