Cone unfolding of space curves 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.
 A: Liberman used cylinder unfolding to study geodesics on convex surfaces.
[Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313.]
Right now standard formulation of Liberman's lemma uses cone unfolding (it gives more general statement and proofs are the same).
Latter Alexandrov used development of curves that could be considered as a generalization of cone unfolding. [A. D. Alexandrow. “Über eine Verallgemeinerung der Riemannschen Geometrie”. Schr. Forschungsinst. Math. 1 (1957), 33–84.]
A: Pardon me for this bit of self-promotion, especially because this is
only tangential to the OP's concerns.
But cone unrolling and Anton Petrunin's mention of Alexandrov's developments, in conjunction with the OP's query---"what other applications of it might be known?"---lead to me to mention that in this paper,

J. O'Rourke and Costin Vîlcu,
"Conical Existence of Closed Curves on Convex Polyhedra."
Computational Geometry, 47, pp.149-163 (2014).

we constructed a curve on a cone that overlaps when developed on the
plane. Avoiding overlap would be useful in several contexts,
but alas...
Here is the relevant figure from that paper.

(a) Open curve on cone of angle $\alpha$, with cone opened.
(b) A different opening of the same cone and curve.
(c) Development of curve self-intersects.
