There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which has been called (cone) unfolding, was used in a paper of Cantarella, Kusner and Sullivan to study thickness of knots, and more recently in a paper with James Wenk to prove Zalgaller's sphere inspection conjecture for closed curves.
I would like to know when this operation may have been first used, and what other applications of it might be known.
The cone unfolding is defined as follows. Let $o$ be a point in $R^n$ not lying on $\gamma$. Consider the conical surface $C$ generated by all the line segments $o\gamma(t)$. The unfolding $\tilde\gamma$ of $\gamma$ with respect to $o$ is the curve obtained by the isometric immersion (or unrolling) of the conical surface $C$ into the plane $R^2$. Note that unfolding preserves the arc length between any pairs of points of $\gamma$, and does not decrease the chord length. In particular, if $\gamma$ is polygonal then $\tilde\gamma$ will be polygonal as well.
If $\gamma$ is closed, i.e., $\gamma(a)=\gamma(b)$ then we may choose the point $o$ so that $\tilde\gamma$ will also be closed. To see this let $\overline\gamma(t):=(\gamma(t)-o)/|\gamma(t)-o|$ be the projection of $\gamma$ into the unit sphere centered at $o$. The total length of $\overline\gamma$ may be called the vision angle $\theta$ of $\gamma$ with respect to $o$. If $o$ lies in the convex hull of $\gamma$, then it follows from Crofton's formula that $\theta\geq 2\pi$. On the other hand $\theta$ may be arbitrarily small if $o$ is chosen far from the convex hull of $\gamma$. So, by continuity, we may choose $o$ so that $\theta=2\pi$. Then $\tilde\gamma$ will close up as desired.
