13,994 research outputs found
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
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
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
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
- …