Unfolding Restricted Convex Caps

TL;DR

This paper presents an algorithm for unfolding a specific class of convex polyhedra called restricted convex caps, using geometric extensions to ensure the unfolded shape does not overlap.

Contribution

It introduces a novel unfolding algorithm for convex caps with quadrilateral faces over integer lattice bases, leveraging extended Cauchy's arm lemma for non-overlap guarantees.

Findings

Algorithm successfully unfolds convex caps without overlaps.

Cap faces are quadrilaterals with vertices on an integer lattice.

The method extends classical geometric lemmas for proof of non-overlap.

Abstract

This paper details an algorithm for unfolding a class of convex polyhedra, where each polyhedron in the class consists of a convex cap over a rectangular base, with several restrictions: the cap's faces are quadrilaterals, with vertices over an underlying integer lattice, and such that the cap convexity is ``radially monotone,'' a type of smoothness constraint. Extensions of Cauchy's arm lemma are used in the proof of non-overlap.