Abstract
We begin with some background on convex polyhedra, setting the context for our results. The discussion in this section will be mostly informal and elementary, with formal definitions and statements deferred to later chapters.
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Notes
- 1.
One may imagine this as a navigation problem on Earth: finding a shortest path between cities. Basic properties of geodesic segments on convex polyhedra are detailed in Sect. 2.1.
- 2.
Details below in Sect. 1.3.
- 3.
We note that this is the area of the unit-sphere polygonal domain determined by the normal cone to P at v.
- 4.
In fact, infinitely many: Theorem 25.1.4 in [DO07].
- 5.
The existence part of AGT holds in a more general setting, which leads to Open Problem 18.7.
- 6.
Formally, a convex surface is the boundary of a convex body in \({\mathbb {R}}^3\) or a doubly covered planar convex body.
- 7.
We will see in Sect. 2.3.2 that U(T) is the star-unfolding with respect to c.
- 8.
Also called a disphenoid, or a tetramonohedron, or an isotetrahedron.
- 9.
In [DO07, Sec. 25.3.1] this structure is called a “rolling belt.”
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O’Rourke, J., Vîlcu, C. (2024). Introduction to Part I. In: Reshaping Convex Polyhedra. Springer, Cham. https://doi.org/10.1007/978-3-031-47511-5_1
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