Related papers: Computing distances on Riemann surfaces

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the…

A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees…

We use the $p$-Laplacian with large $p$-values in order to approximate geodesic distances to features on surfaces. This differs from Fayolle and Belyaev's (2018) [1] computational results using the $p$-Laplacian for the distance-to-surface…

We introduce the notion of log-Riemann surfaces. These are Riemann surfaces given by cutting and pasting planes together isometrically, and come equipped with a holomorphic local diffeomorphism to C called the projection map, and a…

A central aspect of robotic motion planning is collision avoidance, where a multitude of different approaches are currently in use. Optimization-based motion planning is one method, that often heavily relies on distance computations between…

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann…

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…

A novel algorithm is proposed for quantitative comparisons between compact surfaces embedded in the three-dimensional Euclidian space. The key idea is to identify those objects with the associated surface measures and compute a weak…

Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…

In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the…

We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex."…

In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $\Delta_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a…

The constrained minimization (respectively maximization) of directed distances and of related generalized entropies is a fundamental task in information theory as well as in the adjacent fields of statistics, machine learning, artificial…

Many machine learning applications are naturally formulated as optimization problems on Riemannian manifolds. The main idea behind Riemannian optimization is to maintain the feasibility of the variables while moving along a descent…

We revise and extend the algorithm provided in [1] to compute the finite Connes' distance between normal states. The original formula in [1] contains an error and actually only provides a lower bound. The correct expression, which we…

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is…

We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary…

We study the Bergman space interpolation problem of open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. We equip such a surface with what we call an asymptotically flat conformal metric,…

The input to the distant representatives problem is a set of $n$ objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are…

In this paper we study the geometry of metric spheres in the curve complex of a surface, with the goal of determining the "average" distance between points on a given sphere. Averaging is not technically possible because metric spheres in…