eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-06-23
9111
1
18
10.4230/DagSemProc.09111.1
article
09111 Abstracts Collection – Computational Geometry
Agarwal, Pankaj Kumar
Alt, Helmut
Teillaud, Monique
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol09111/DagSemProc.09111.1/DagSemProc.09111.1.pdf
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-06-23
9111
1
22
10.4230/DagSemProc.09111.2
article
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Kane, Daniel
Price, Gregory Nathan
Demaine, Erik
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol09111/DagSemProc.09111.2/DagSemProc.09111.2.pdf
Folding
metrics
pseudopolynomial
algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-06-23
9111
1
16
10.4230/DagSemProc.09111.3
article
Minimizing Absolute Gaussian Curvature Locally
Giesen, Joachim
Madhusudan, Manjunath
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol09111/DagSemProc.09111.3/DagSemProc.09111.3.pdf
Absolute Gaussian curvature
surface reconstruction
mesh smoothing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-06-23
9111
1
3
10.4230/DagSemProc.09111.4
article
Open Problem Session
Mitchell, Joseph S.
This is a scribing of the open problems posed at the Tuesday evening open problem session. Posers of problems provided input after the session.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol09111/DagSemProc.09111.4/DagSemProc.09111.4.pdf
Open problems
computational geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-06-23
9111
1
30
10.4230/DagSemProc.09111.5
article
Shortest Path Problems on a Polyhedral Surface
Wenk, Carola
Cook, Atlas F.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol09111/DagSemProc.09111.5/DagSemProc.09111.5.pdf
Shortest paths
edge sequences
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2009-06-23
9111
1
3
10.4230/DagSemProc.09111.6
article
Two Applications of Point Matching
Rote, Günter
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)
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol09111/DagSemProc.09111.6/DagSemProc.09111.6.pdf
Bipartite matching
least-squares