TITLE: Finding maximal prefix of a simple curve
QUESTION [7 upvotes]: Let $S$ be a piecewise-linear simple curve. For a point $p_1$ on $S$ I want to determine a maximal interval $I$ on $S$ starting in $p_1$ such that $I$ is contained in a unit disk. Is this possible, or has it perhaps already been solved in the past, and I am just unable to find an answer?
I thought of something like the following: Place points uniformly spaced along $S$ and greedily include as many points as possible while the minimum enclosing disk is at most the unit disk. Assume that $p_{k+1}$ is a point on $S$ that cannot be included as the minimum enclosing disk  containing $p_1, p_2,...,p_{k+1}$ would become larger than the unit disk. A unit disk would be able to cover points $p_1, p_2,...,p_{k}$ as well as some segment of the curve between $p_k$ and $p_{k+1}$, however actually determining this seems non-trivial.
Edit: I have noted that the boundary of the unit disk containing $I$ need not intersect $p_1$. See the figure below. Here the black simple curve is $S$, the orange circle is a unit disk and the blue interval on $S$ is the maximal interval $I$. This seems useful, but I cannot quite tell why yet though.

REPLY [3 votes]: 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.