Unfolding Restricted Convex Caps

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

On the Development of the Intersection of a Plane With a Polytope

Define a “slice” curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy\u27s arm lemma to permit nonconvex “openings” of a planar convex chain

Edge-Unfolding Nearly Flat Convex Caps

The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in ℝ3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. Nearly flat means that every outer face normal forms a sufficiently small angle φ \u3c Φ with the z-axis orthogonal to the halfspace bounding plane. The size of Φ depends on the acuteness gap α: if every triangle angle is at most π/2 - α, then Φ ≈ 0.36√α suffices; e.g., for α = 3°, Φ ≈ 5°. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n); a version has been implemented

Some Properties of Yao Y\u3csub\u3e4\u3c/sub\u3e Subgraphs

The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners

The Yao Graph Y\u3csub\u3e6\u3c/sub\u3e is a Spanner

We prove that Y6 is a spanner. Y6 is the Yao graph on a set of planar points, which has an edge from each point x to a closest point y within each of the six angular cones of 60◦ surrounding x

Unfolding Convex Polyhedra via Radially Monotone Cut Trees

A notion of radially monotone cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does not provably find a radially monotone cut tree, it in fact does find such a tree with high frequency, and after cutting unfolds without overlap. This performance of a greedy algorithm leads to the conjecture that spherical polyhedra always have a radially monotone cut tree and unfold without overlap

Computational Geometry Column 32

The proof of Dey\u27s new k-set bound is illustrated

Spiral Unfoldings of Convex Polyhedra

The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution

Computational Geometry Column 32

The proof of Dey\u27s new k-set bound is illustrated

Hypercube Unfoldings that Tile R\u3csup\u3e3\u3c/sup\u3e and R\u3csup\u3e2\u3c/sup\u3e

We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a dimension-descending tiler. We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane