Finding maximal prefix of a simple curve Let $S$ be a simple curve. I want to determine maximal prefix of $S$ contained in a unit circle. Is this possible, or has it perhaps already been solved in the past, and I am just unable to find an answer?
 A: I haven't worked this out carefully, but here's an approach.
Use J.J.Green's nice idea to identify the last partially covered segment $p_k p_{k+1}$.
Let $H_k$ be the convex hull of $p_1,\ldots,p_k$. Clearly covering $H_k$
covers the segments from $p_1$ to $p_k$.
Any pair of vertices of $H_k$ determine a unit disk through those two points. Some of these enclose all of $H_k$, some do not. Among those that
include $H_k$, record how much of $p_k p_{k+1}$ is captured by each unit disk. In the image below, the red circle is the winner.
     
But there remains the possibility that the unit disk touches only one
vertex $p_i$ of $H_k$, and is free to rotate about $p_i$ until it bangs
into $H_k$ (when it then touches two points). So in that interval of rotation, it encloses all of $H_k$ and covers a portion of $p_k p_{k+1}$.
So now we have reduced the problem to a unit-circle through one point $p_i$,
and as it rotates about $p_i$, computing where the circle
intersects $p_k p_{k+1}$. This is the part I haven't worked out, but
it is a calculation with one variable $\phi$, the rotation about $p_i$,
and it should not be too complicated to find the maximum over a range
of $\phi$.
The overall idea is to "walk" or roll the unit circle around $H_k$, recording
the maximum extent of that last edge captured.
