Delaunay triangulations of generalized Bolza surfaces

Ebbens, Matthijs; Iordanov, Iordan; Teillaud, Monique; Vegter, Gert

doi:10.20382/jocg.v13i1a5

Cited by 3 publications

(4 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we apply our general results to the generalized Bolza surfaces [25], highly symmetric surfaces of any genus g ⩾ 2. We denote them by S g .…”

Section: Application To Generalized Bolza Surfacesmentioning

confidence: 99%

See 1 more Smart Citation

Computing distances on Riemann surfaces

Stepanyants,

Beardon,

Paton

et al. 2024

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1.

show abstract

“…In this section, we apply our general results to the generalized Bolza surfaces [25], highly symmetric surfaces of any genus g ⩾ 2. We denote them by S g .…”

Section: Application To Generalized Bolza Surfacesmentioning

confidence: 99%

“…We also develop a fast algorithm for computing distances on these surfaces using a different set of ideas. For illustrative purposes, we show how the formula and the algorithm can be applied to computing distances on generalized Bolza surfaces [25]. These are highly symmetric hyperbolic Riemann surfaces of any genus g ⩾ 2.…”

Section: Introductionmentioning

confidence: 99%

Computing distances on Riemann surfaces

Stepanyants,

Beardon,

Paton

et al. 2024

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…By Poincaré's theorem [1], the interior of the polygon is a fundamental domain of S g , while the polygon itself is called the fundamental polygon. The maximally symmetric surface S 2 of genus 2 is the well-known Bolza surface [4], while the maximally symmetric surfaces with g > 2 are generalized Bolza surfaces [5].…”

Section: Background Information and Definitionsmentioning

confidence: 99%

“…The definition of maximally symmetric S g is in the next section. In particular, S 2 is known as the Bolza surface [4], while S g with g > 1 are generalized Bolza surfaces [5]. A free particle moving along a geodesic on the Bolza surface was the first dynamical system proven rigorously to be chaotic [6].…”

Section: Introductionmentioning

confidence: 99%

Diameter of Compact Riemann Surfaces

Stepanyants¹,

Beardon²,

Paton³

et al. 2023

Preprint

View full text Add to dashboard Cite

Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple closed-form expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameter of the maximally symmetric Riemann surface of any genus greater than 1 is equal to the radius of its fundamental polygon.

show abstract

Diameter of Compact Riemann Surfaces

Stepanyants,

Beardon,

Paton

et al. 2024

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold.

show abstract

Delaunay triangulations of generalized Bolza surfaces

Cited by 3 publications

References 24 publications

Computing distances on Riemann surfaces

Computing distances on Riemann surfaces

Diameter of Compact Riemann Surfaces

Diameter of Compact Riemann Surfaces

Contact Info

Product

Resources

About