arXiv research

New theorem proves convergence of various discrete conformal structures to conformal maps.

The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.

Discrete conformal maps on surfaces with vertex decorations are studied.

We establish a connection between recent developments in the study of vortices in the abelian Higgs models, and in the theory of structure-preserving discrete conformal maps. We explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form, and sho…

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 s…

Liouville's theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding …

Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …

Discrete approximation solves Björling's minimal surface problem.

A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some mar…

Method computes harmonic and conformal maps from point clouds.

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circ…

Paper proves circle packings converge to Riemann mapping for Jordan domains.

In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $Ω$ and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ denote a triangulation of $Ω\cup\bordΩ$. We construct a \emph{new} decomposition of $Ω\cup\bo…

The paper studies dual pairs of generic conformally flat hypersurfaces in 4-space.

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e.…

Plane triangulations remain rigid under discrete conformal changes.

The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.

Plane Delaunay triangulations are rigid under Luo's discrete conformal change.

Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.

This paper completes the classification of discrete conformal structures on surfaces.

Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.

Let $M$ be an open Riemann surface and let $Λ\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $Λ$ and whose generalized Gauss map $M\to\mathbb{CP}^{n-1}$, $n\ge 3$, avoids $n$ hyperplanes of $\m…

In recent decades, the use of 3D point clouds has been widespread in computer industry. The development of techniques in analyzing point clouds is increasingly important. In particular, mapping of point clouds has been a challenging problem. In this paper, we develop a discrete analogue of the Teichmüller extremal mapp…

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …

Study on deforming discrete conformal structures on surfaces with boundaries.

Study on discrete Gaussian curvature for polyhedral surfaces.

New method approximates Gaussian curvature on discrete surfaces.

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this pr…

The paper proves rigidity and ergodicity of horospherical foliations.

This paper classifies discrete conformal structures on surfaces with boundary.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.

We study surfaces with decorations and prove uniformization in non-Euclidean geometries.

Paper proves rigidity of discrete conformal structures on polyhedral surfaces.

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of…

Study shows how to detect representation extendability using conformal measures.

Unique metric found for discrete curvature on spherical cone-metrics.

Fractional combinatorial flow improves surface conformal structures.

A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Ad…

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

New boundary and point constraints for controlling conformal surfaces.

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…

New interpretation of discrete conformality using polyhedral convex hulls.

Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the…