38 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
The worst approximable rational numbers
We classify and enumerate all rational numbers with approximation constant at
least using hyperbolic geometry. Rational numbers correspond to
geodesics in the modular torus with both ends in the cusp, and the
approximation constant measures how far they stay out of the cusp neighborhood
in between. Compared to the original approach, the geometric point of view
eliminates the need to discuss the intricate symbolic dynamics of continued
fraction representations, and it clarifies the distinction between the two
types of worst approximable rationals: (1) There is a plane forest of Markov
fractions whose denominators are Markov numbers. They correspond to simple
geodesics in the modular torus with both ends in the cusp. (2) For each Markov
fraction, there are two infinite sequences of companions, which correspond to
non-simple geodesics with both ends in the cusp that do not intersect a pair of
disjoint simple geodesics, one with both ends in the cusp and one closed.Comment: 50 pages, 18 figures. v2: Flahive and Gurwood references added;
abstract, introduction, and acknowledgements update
Discrete Yamabe problem for polyhedral surfaces
We introduce a new discretization of the Gaussian curvature on piecewise at surfaces. As the prime new feature the curvature is scaled by the factor 1/r2 upon scaling the metric globally with the factor r. We develop a variational principle to tackle the corresponding discrete uniformisation theorem – we show that each piecewise at surface is discrete conformally equivalent to one with constant discrete Gaussian curvature. This surface is in general not unique. We demonstrate uniqueness for particular cases and disprove it in general by providing explicit counterexamples. Special attention is paid to dealing with change of combinatorics
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
Hyperbolic constant mean curvature one surfaces: Spinor representation and trinoids in hypergeometric functions
We present a global representation for surfaces in 3-dimensional hyperbolic
space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic
spinors. This is a modification of Bryant's representation.
It is used to derive explicit formulas in hypergeometric functions for CMC-1
surfaces of genus 0 with three regular ends which are asymptotic to catenoid
cousins (CMC-1 trinoids).Comment: 29 pages, 9 figures. v2: figures of cmc1-surfaces correcte
A unique representation of polyhedral types
It is known that for each combinatorial type of convex 3-dimensional
polyhedra, there is a representative with edges tangent to the unit sphere.
This representative is unique up to projective transformations that fix the
unit sphere.
We show that there is a unique representative (up to congruence) with edges
tangent to the unit sphere such that the origin is the barycenter of the points
where the edges touch the sphere.Comment: 4 pages, 2 figures. v2: belated upload of final version (of March
2004