Abstract
In this chapter we briefly present basic properties of geodesics and cut loci on convex polyhedra, the star-unfolding, prove a rigidity result, and describe the technique of vertex-merging. All of these geometric tools will be needed subsequently. The reader might skip this section and return to it as the tools are deployed, for which purpose we provide pointers.
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Notes
- 1.
That is, around each point.
- 2.
The proper statement of Gauss–Bonnet Theorem concerns smooth surfaces. This discrete variation is generally credited much earlier to René Descartes.
- 3.
See, e.g., [DO11][Thm. 6.29].
- 4.
When defined equivalently as the locus of all cut points, the cut locus can be found useful in more general frameworks, i.e., where the notion of geodesic segment makes sense.
- 5.
In some literature, these points are called “branch points” or “junctions” of \({\mathcal C}(x)\).
- 6.
Unlike the cut locus, which generalizes to \({\mathbb {R}}^d\), d > 3, the star-unfolding has no analog in higher dimensions [MP08].
- 7.
The triangle inequality holds by definition in every metric space, hence also for geodesic triangles.
- 8.
As mentioned in Chap. 1, intrinsic means independent of the embedding in \({\mathbb {R}}^3\).
- 9.
An open intrinsic disk of radius r > 0 is the set of points at intrinsic distance less than r from some fixed point.
- 10.
See the discussion in Sect. 2.3.2.
- 11.
Notice that γ need not be an edge of P nor even a geodesic segment between v1, v2. Since γ is a geodesic arc, it passes through no vertex of P, and hence it develops in the plane to a line segment.
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O’Rourke, J., Vîlcu, C. (2024). Preliminaries. In: Reshaping Convex Polyhedra. Springer, Cham. https://doi.org/10.1007/978-3-031-47511-5_2
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