I learned of the paper "Closing curves by rearranging arcs" by L. Alese (arXiv link) by a post on Reddit and was curious about related questions and generalizations. Here a rearrangement of a curve is where you cut the curve into finitely many arcs and then through rigid plane motions glue the arcs back together so that the tangents at the endpoints match.
What does the space of all rearrangements of a curve look like? For straight line segments and circles you get nothing: the curve is the same no matter how you cut and rearrange pieces. For curves with non-constant curvature the possibilities increase. The paper above answers the question "Does the rearrangement space contain a closed curve?"
My guess is the space is determined by the curvature measure along the curve. This is a signed measure whose integral gives the turning number along the curve; it is possibly singular if there are any sharp turns.
It seems like the "boundary" or "completion" of the rearrangement space is determined by the total variation of the curvature measure in some sense, I'm not sure how to make this precise.
Added on:
Say you have a finite signed curvature measure $\kappa$, defined on the Borel sets of $I=[0,1]$. The corresponding angle function is $\theta(t)=\kappa([0,t])$ and the corresponding curve is
$$x(t)=\int_0^t\cos(\theta(s))ds$$
$$y(t)=\int_0^t\sin(\theta(s))ds.$$ A rearrangement of $\kappa$ is determined by a partition of $I$ into sub intervals $I_1=(0,t_1],I_2=(t_1,t_2],\ldots,I_k=(t_{k-1},1]$ together with a permutation $\sigma$ of $\{1,\ldots,k\}$.
This gives a map $P:I\to I$ defined by rearranging the $I_j$ according to $\sigma$ and then stacking them back over the interval $I$, together with saying $P(0)=0$. For example, with the partition $I_1=(0,\frac{3}{4}], I_2=(\frac{3}{4},1]$ and the (cycle notation) permutation $(1 2)$: $$P(t)=\begin{cases}t+\frac{3}{4}, \text{ if } 0<t\leq\frac{1}{4}\\t-\frac{1}{4}, \text{ if } \frac{1}{4}<t\leq 1\\ 0, \text{ if } t=0\end{cases}.$$
The rearranged curvature measure is defined by
$$\tilde\kappa(A)=\kappa(P(A)),$$
and the space of all rearrangements is given by applying this construction over all partitions and all permutations.
Starting from a given $\kappa$, what does the space of rearrangements look like?
