Dagstuhl Seminar Proceedings, Volume 9111
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
9111
2009
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-9111
09111 Abstracts Collection – Computational Geometry
From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 ``Computational Geometry '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
1-18
Regular Paper
Pankaj Kumar
Agarwal
Pankaj Kumar Agarwal
Helmut
Alt
Helmut Alt
Monique
Teillaud
Monique Teillaud
10.4230/DagSemProc.09111.1
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of some convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution is the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron given the metric, and prove a pseudopolynomial bound on its running time.
Folding
metrics
pseudopolynomial
algorithms
1-22
Regular Paper
Daniel
Kane
Daniel Kane
Gregory Nathan
Price
Gregory Nathan Price
Erik
Demaine
Erik Demaine
10.4230/DagSemProc.09111.2
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Minimizing Absolute Gaussian Curvature Locally
One of the remaining challenges when reconstructing a surface from a
finite sample is recovering non-smooth surface features like sharp
edges. There is practical evidence showing that a two step approach
could be an aid to this problem, namely, first computing a
polyhedral reconstruction isotopic to the sampled surface, and
secondly minimizing the absolute Gaussian curvature of this
reconstruction globally. The first step ensures topological
correctness and the second step improves the geometric accuracy of
the reconstruction in the presence of sharp features without
changing its topology. Unfortunately it is computationally hard to
minimize the absolute Gaussian curvature globally. Hence we study a
local variant of absolute Gaussian curvature minimization problem
which is still meaningful in the context of surface
fairing. Absolute Gaussian curvature like Gaussian curvature is
concentrated at the vertices of a polyhedral surface embedded into
$mathbb{R}^3$. Local optimization tries to move a single vertex in
space such that the absolute Gaussian curvature at this vertex is
minimized. We show that in general it is algebraically hard to find
the optimal position of a vertex. By algebraically hard we mean that
in general an optimal solution is not constructible, i.e., there
exist no finite sequence of expressions starting with rational numbers,
where each expression is either the sum, difference,
product, quotient or $k$'th root of preceding expressions and the
last expressions give the coordinates of an optimal solution. Hence
the only option left is to approximate the optimal position. We
provide an approximation scheme for the minimum possible value of
the absolute Gaussian curvature at a vertex.
Absolute Gaussian curvature
surface reconstruction
mesh smoothing
1-16
Regular Paper
Joachim
Giesen
Joachim Giesen
Manjunath
Madhusudan
Manjunath Madhusudan
10.4230/DagSemProc.09111.3
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Open Problem Session
This is a scribing of the open problems posed at the Tuesday evening open problem session. Posers of problems provided input after the session.
Open problems
computational geometry
1-3
Regular Paper
Joseph S.
Mitchell
Joseph S. Mitchell
10.4230/DagSemProc.09111.4
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Shortest Path Problems on a Polyhedral Surface
We develop algorithms to compute edge sequences, Voronoi diagrams, shortest
path maps, the Fréchet distance, and the diameter for a polyhedral surface. Distances on the surface are measured either by the length of a Euclidean shortest path or by link distance. Our main result is a linear-factor speedup for computing all shortest path edge sequences on a convex polyhedral surface.
Shortest paths
edge sequences
1-30
Regular Paper
Carola
Wenk
Carola Wenk
Atlas F.
Cook
Atlas F. Cook
10.4230/DagSemProc.09111.5
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Two Applications of Point Matching
The two following problems can be solved by a reduction
to a minimum-weight bipartite matching problem (or a related
network flow problem):
a) Floodlight illumination:
We are given $n$ infinite wedges (sectors, spotlights) that can cover
the whole plane when placed at the origin.
They are to be assigned to $n$ given locations
(in arbitrary order, but without rotation)
such that they still cover the whole plane.
(This extends results of Bose et al. from 1997.)
b) Convex partition:
Partition a convex $m$-gon into $m$ convex parts, each part
containing one of the edges and a given number of points from a given
point set. (Garcia and Tejel 1995, Aurenhammer 2008)
Bipartite matching
least-squares
1-3
Regular Paper
Günter
Rote
Günter Rote
10.4230/DagSemProc.09111.6
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode