arXiv subjects

Carl O. R. Lutz

Publications and source records attributed to Carl O. R. Lutz.

5 recordsLinked to original sources

arXiv2026

Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings

Hyperbolic inversive distance circle packings on the $2$-sphere correspond to Koebe polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic $3$-space. Koebe polyhedra are triangulated convex hyperbolic polyhedra with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. We prove the global rigidity of these circle packings or, equivalently, of these Koebe polyhedra under mild assumptions on the links of their vertices. Previous rigidity results apply only when all edges of the Koebe polyhedron are tangent or, alternatively, when no edge is tangent to the ideal boundary of hyperbolic space. We remove these restrictions. This generalizes the global rigidity results of both Bao-Bonahon and Bowers-Bowers-Pratt (arXiv:1703.09338), as well as the uniqueness part of the celebrated Koebe-Andre'ev-Thurston Theorem to the case where adjacent circles need not touch.

math.MG↗

arXiv2023

Decorated discrete conformal equivalence in non-Euclidean geometries

We introduce decorated piecewise hyperbolic and spherical surfaces and discuss their discrete conformal equivalence. A decoration is a choice of circle about each vertex of the surface. Our decorated surfaces are closely related to inversive distance circle packings, canonical tessellations of hyperbolic surfaces, and hyperbolic polyhedra. We prove the corresponding uniformization theorem. Furthermore, we show that on can deform continuously between decorated piecewise hyperbolic, Euclidean, and spherical surfaces sharing the same fundamental discrete conformal invariant. Therefore, there is one master theory of discrete conformal equivalence in different background geometries. Our approach is based on a variational principle, which also provides a way to compute the discrete uniformization and geometric transitions.

math.GT↗

arXiv2023

Decorated discrete conformal maps and convex polyhedral cusps

We discuss a notion of discrete conformal equivalence for decorated piecewise euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. Under the assumption that the circles are non-intersecting, we proof the corresponding discrete uniformization theorem. The uniformization theorem for discrete conformal maps corresponds to the special case that all circles degenerate to points. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. It is based on a concave variational principle, which also provides a method for the computation of decorated discrete conformal maps.

math.GT↗

arXiv2022

Canonical tessellations of decorated hyperbolic surfaces

A decoration of a hyperbolic surface of finite type is a choice of circle, horocycle or hypercycle about each cone-point, cusp or flare of the surface, respectively. In this article we show that a decoration induces a unique canonical tessellation and dual decomposition of the underlying surface. They are analogues of the weighted Delaunay tessellation and Voronoi decomposition in the Euclidean plane. We develop a characterisation in terms of the hyperbolic geometric equivalents of Delaunay's empty-discs and Laguerre's tangent-distance, also known as power-distance. Furthermore, the relation between the tessellations and convex hulls in Minkowski space is presented, generalising the Epstein-Penner convex hull construction. This relation allows us to extend Weeks' flip algorithm to the case of decorated finite type hyperbolic surfaces. Finally, we give a simple description of the configuration space of decorations and show that any fixed hyperbolic surface only admits a finite number of combinatorially different canonical tessellations.

math.GT↗

arXiv2020

Non-Euclidean Laguerre geometry and incircular nets

Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hyperspheres, and their oriented contact in Euclidean space. We describe how this can be generalized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study the corresponding groups of Laguerre transformations. We give an introduction to Lie geometry and describe how these Laguerre geometries can be obtained as subgeometries. As an application of two-dimensional Lie and Laguerre geometry we study the properties of incircular nets.

math.MG↗