As an example of variation of gradient, let's consider encoding of shapes.
Shape of a trajectory could ...
And the hexagon on the left is encoded into the binary sequence of length 10 on the right, using the trajectory shown below. Shown below is the example we considered before: the closed trajectory defined by three peaks.
First, the gradient of the first two flat tiles t[0] and t[1] are given as shown in blue.
Suppose that the "2nd derivative" of t[0] is -1. Then, since the first two tiles have the same gradient, the "2nd derivative" of t[1] is also -1. When the gradient changes over t[3], the "2nd derivative" of t[3] becomes +1 accordingly.
Note that there is a one-to-one correspondence between the "2nd derivative" and "up and down" along trajectory.