Pardon me for this bit of self-promotion, especially because this is only a tangent to the OP's question. 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.

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.

Pardon me for this bit of self-promotion, especially because this is only a tangent to the OP's question. But cone unrolling and Anton Petrunin's mention of Alexandrov's developments  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.

Pardon me for this bit of self-promotion, especially because this is only a tangent to the OP's question. 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.