18 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
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
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow
Modelling incompressible ideal fluids as a finite collection of vortex
filaments is important in physics (super-fluidity, models for the onset of
turbulence) as well as for numerical algorithms used in computer graphics for
the real time simulation of smoke. Here we introduce a time-discrete evolution
equation for arbitrary closed polygons in 3-space that is a discretisation of
the localised induction approximation of filament motion. This discretisation
shares with its continuum limit the property that it is a completely integrable
system. We apply this polygon evolution to a significant improvement of the
numerical algorithms used in Computer Graphics.Comment: 15 pages, 3 figure
A discrete Laplace-Beltrami operator for simplicial surfaces
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It
depends only on the intrinsic geometry of the surface and its edge weights are
positive. Our Laplace operator is similar to the well known finite-elements
Laplacian (the so called ``cotan formula'') except that it is based on the
intrinsic Delaunay triangulation of the simplicial surface. This leads to new
definitions of discrete harmonic functions, discrete mean curvature, and
discrete minimal surfaces. The definition of the discrete Laplace-Beltrami
operator depends on the existence and uniqueness of Delaunay tessellations in
piecewise flat surfaces. While the existence is known, we prove the uniqueness.
Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides
an optimality criterion for Delaunay triangulations, and this can be used to
prove that the edge flipping algorithm terminates also in the setting of
piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay
triangulations of piecewise flat surfaces revised and expanded. References
added. Some minor changes, typos corrected. v3: fixed inaccuracies in
discussion of flip algorithm, corrected attributions, added references, some
minor revision to improve expositio
There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems
There is no 5,7-triangulation of the torus, that is, no triangulation with
exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no
3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be
bicolored. Similar statements hold for 4,8-triangulations and
2,6-quadrangulations. We prove these results, of which the first two are known
and the others seem to be new, as corollaries of a theorem on the holonomy
group of a euclidean cone metric on the torus with just two cone points. We
provide two proofs of this theorem: One argument is metric in nature, the other
relies on the induced conformal structure and proceeds by invoking the residue
theorem. Similar methods can be used to prove a theorem of Dress on infinite
triangulations of the plane with exactly two irregular vertices. The
non-existence results for torus decompositions provide infinite families of
graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to
appear in Geometriae Dedicat