Let $f\colon [0,1]\to \mathbf R^n$ and $g\colon[0,1]\to\mathbf R^n$ parametrize your curves. The set $C$ of $t\in[0,1]$ such that there exists $s\in[0,1]$ with $f(t)=g(s)$ is a closed subset of $[0,1]$. More importantly, it is definable in the sense of model theory in the language — called $\mathbf R_{\text{an}}$ — consisting of polynomial functions and so called restricted analytic functions, namely analytic functions defined over compact subsets of $\mathbf R^m$.

It is an important theorem (Gabrielov; Denef and van den Dries) that $\mathbf R_{\text{an}}$ is $o$-minimal. This means that definable subsets of the real line are finite unions of intervals. 

From there, one should be able to analyse the situation further when the coincidence set is infinite. I would conjecture that it can only be the union of one or two intervals. Ramiro gave an example with one interval, though more complicated examples are possible, e.g., of the form $s\mapsto F(u(s))$, $s\mapsto F(v(s))$, where $u,v\colon[0,1]\to\mathbf R$ are analytic and $F\colon\mathbf R\to\mathbf R^n$ is a fixed curve.  In particular, the set is a union of two intervals if $F$ is a parameterization of a circle and $u$ and $v$ are suitably chosen so as to draw two arcs which overlap twice.

Two different analytic curves cannot intersect in infinitely many points
Is there a way to check which server OS a particular site is running. For example from linux command line. Maybe via telnet? But all I can get from "telnet www.google.com http" is:

Trying 209.85.173.104...
Connected to www.l.google.com.
Escape character is '^]'.


Can somebody dwell on the subject if possible at all?
