1,338 research outputs found
Variational principles for circle patterns
A Delaunay cell decomposition of a surface with constant curvature gives rise
to a circle pattern, consisting of the circles which are circumscribed to the
facets. We treat the problem whether there exists a Delaunay cell decomposition
for a given (topological) cell decomposition and given intersection angles of
the circles, whether it is unique and how it may be constructed. Somewhat more
generally, we allow cone-like singularities in the centers and intersection
points of the circles. We prove existence and uniqueness theorems for the
solution of the circle pattern problem using a variational principle. The
functionals (one for the euclidean, one for the hyperbolic case) are convex
functions of the radii of the circles. The analogous functional for the
spherical case is not convex, hence this case is treated by stereographic
projection to the plane. From the existence and uniqueness of circle patterns
in the sphere, we derive a strengthened version of Steinitz' theorem on the
geometric realizability of abstract polyhedra.
We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and
Rivin for circle packings and circle patterns from our variational principles.
In the case of Br\"agger's and Rivin's functionals. Leibon's functional for
hyperbolic circle patterns cannot be derived directly from our functionals. But
we construct yet another functional from which both Leibon's and our
functionals can be derived.
We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics
Gauss images of hyperbolic cusps with convex polyhedral boundary
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral
boundary is uniquely determined by its Gauss image. Furthermore, any spherical
metric on the torus with cone singularities of negative curvature and all
closed contractible geodesics of length greater than is the metric of
the Gauss image of some convex polyhedral cusp. This result is an analog of the
Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in
terms of their Gauss images.
The proof uses a variational method. Namely, a cusp with a given Gauss image
is identified with a critical point of a functional on the space of cusps with
cone-type singularities along a family of half-lines. The functional is shown
to be concave and to attain maximum at an interior point of its domain. As a
byproduct, we prove rigidity statements with respect to the Gauss image for
cusps with or without cone-type singularities.
In a special case, our theorem is equivalent to existence of a circle pattern
on the torus, with prescribed combinatorics and intersection angles. This is
the genus one case of a theorem by Thurston. In fact, our theorem extends
Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's
theorem on compact convex polyhedra with non-obtuse dihedral angles.
The functional used in the proof is the sum of a volume term and curvature
term. We show that, in the situation of Thurston's theorem, it is the potential
for the combinatorial Ricci flow considered by Chow and Luo.
Our theorem represents the last special case of a general statement about
isometric immersions of compact surfaces.Comment: 55 pages, 17 figure
Minimal surfaces from circle patterns: Geometry from combinatorics
We suggest a new definition for discrete minimal surfaces in terms of sphere
packings with orthogonally intersecting circles. These discrete minimal
surfaces can be constructed from Schramm's circle patterns. We present a
variational principle which allows us to construct discrete analogues of some
classical minimal surfaces. The data used for the construction are purely
combinatorial--the combinatorics of the curvature line pattern. A
Weierstrass-type representation and an associated family are derived. We show
the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended
introduction. Minor changes in presentation. v3: revision according to the
referee's suggestions, improved & expanded exposition, references added,
minor mistakes correcte
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 a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
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