Geometric Realization of Möbius Triangulations

A Möbius triangulation is a triangulation on the Möbius band. A geometric realization of a map M on a surface $\Sigma$ is an embedding of $\Sigma$ into a Euclidean 3-space $\mathbb{R}^3$ such that each face of M is a flat polygon. In this paper, we shall prove that every 5-connected triangulation on the Möbius band has a geometric realization. In order to prove it, we prove that if G is a 5-connected triangulation on the projective plane, then for any face f of G, the Möbius triangulation $G-f$ obtained from G by removing the interior of f has a geometric realization

Universality theorems for inscribed polytopes and Delaunay triangulations

We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to M\"obius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mn\"ev universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, our results imply that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure

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

On a new conformal functional for simplicial surfaces

We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing the squared local conformal discrete Willmore energy $W$ this functional also minimizes local differences of the angles $\beta$. We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are observed. In particular for most of randomly generated simplicial polyhedra the minimizers of $W_2$ and $W_{2,w}$ are inscribed polyhedra. We demonstrate also some applications in geometry processing, for example, a conformal deformation of surfaces to the round sphere. A partial theoretical explanation through quadratic optimization theory of some observed phenomena is presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and Surfaces, 8th International Conference", June 201

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.Comment: 41 pages, 10 figures. arXiv admin note: text overlap with arXiv:2305.1098

Polytopality of simple games

The Bier sphere $Bier(\mathcal{G}) = Bier(K) = K\ast_\Delta K^\circ$ and the canonical fan $Fan(\Gamma) = Fan(K)$ are combinatorial/geometric companions of a simple game $\mathcal{G} = (P,\Gamma)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $\Gamma\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\setminus \Gamma$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $\Gamma$ such that $Bier(\mathcal{G})$ (respectively $Fan(\Gamma)$) is canonically polytopal (canonically pseudo-polytopal) and show, by an experimental/theoretical argument, that all simple games with at most five players are polytopal