I'd like to capture the intuitive notion that a Jordan curve $\gamma_2$ “follows” or “approximates” another Jordan curve $\gamma_1$, i.e. goes somehow “parallel” to it or “oscillates” around it.

Consider a differentiable Jordan curve $\gamma_1: [0,1] \rightarrow \mathbb{R}^2$ and its normals, seen as straight lines crossing the curve perpendicularly. Consider another Jordan curve $\gamma_2$ with the following properties:

Each normal of $\gamma_1$ crosses $\gamma_2$ at least once. I.e., for each normal $n(s_1)$ of $\gamma_1$ there is an $s_2$ such that the point $\gamma_2(s_2)$ lies on $n(s_1)$.

Furthermore when $s_1 < s_1'$ then there are unique $s_2 \le s_2'$ such that $\gamma_2(s_2)$ lies on $n(s_1)$ and $\gamma_2(s_2')$ lies on $n(s_1')$.

Do these conditions suffice to capture the notion described above? For which “pathological ” cases do they eventually fail? How then would they have to be adjusted to capture the notion?

Further questions:

Under which extra conditions does “$\gamma_2$ follows $\gamma_1$” imply that $\gamma_1$ follows $\gamma_2$?

When $\gamma_2$ follows $\gamma_1$, (how) can the area enclosed by $\gamma_1$ and $\gamma_2$ be calculated via the distance function $d(s_1) = |\gamma_1(s_1) - \gamma_2(s_2)|$

($s_2$ the unique parameter according to condition 2 above)?

(Let the area enclosed by $\gamma_1$ and $\gamma_2$ be the symmetric difference $X_1 \triangle X_2 = (X_1 \cup X_2) \setminus (X_1 \cap X_2)$ of the areas $X_1, X_2$ enclosed by $\gamma_1$ and $\gamma_2$.)

Fréchet distancemight be an appropriate measure? Informally it is the shortest leash that allow a master walking on one curve to walk a dog who follows the other curve. en.wikipedia.org/wiki/Fr%C3%A9chet_distance $\endgroup$ – Joseph O'Rourke Jan 7 '13 at 1:18