Computational Geometry Column 35

The subquadratic algorithm of Kapoor for finding shortest paths on a polyhedron is described

Band Unfoldings and Prismatoids: A Counterexample

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.Comment: 5 pages, 3 figures. v2 replaced Fig.1(b) and Fig.3 to illustrate the angles delta=(1/2)epsilon (rather than delta=epsilon

On Folding a Polygon to a Polyhedron

We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier.Comment: 6 pages, 1 figur

Flat Zipper-Unfolding Pairs for Platonic Solids

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.Comment: 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfolding

Computational Geometry Column 41

The recent result that n congruent balls in R^d have at most 4 distinct geometric permutations is described.Comment: To appear in SIGACT News and in Internat. J. Comput. Geom. App

Computational Geometry Column 45

The algorithm of Edelsbrunner for surface reconstruction by ``wrapping'' a set of points in R^3 is described.Comment: 4 pages, to appear in SIGACT News and in IJCGA, 200

Computational Geometry Column 43

The concept of pointed pseudo-triangulations is defined and a few of its applications described.Comment: 3 pages, 1 figur