Hyper-ideal Circle Patterns with Cone Singularities

Abstract

The main objective of this study is to understand how geometric hyper-ideal circle patterns can be constructed from given combinatorial angle data. We design a hybrid method consisting of a topological/deformation approach augmented with a variational principle. In this way, together with the question of characterization of hyper-ideal patterns in terms of angle data, we address their constructability via convex optimization. We presents a new proof of the main results from Jean-Marc Schlenker's work on hyper-ideal circle patterns by developing an approach that is potentially more suitable for applications.Intuitively speaking, a hyper-ideal circle pattern on a surface S is a surface homeomorphic to S, obtained by gluing together decorated geodesic polygons along pairs of corresponding edges. The edges that are being identified should have the same length, the identification should be an isometry and the vertices that get identified should have vertex-circles with same radii.Observe that a hyper-ideal circle pattern on S consists of (i) a cone-metric d on S, (ii) a set of vertices V ⊇ sing(d), (iii) an assignment of vertex radii r on V , and (iv) a geodesic cell complex C d together with (v) a collection of vertex circles and (vi) a collection of face circles. However, the geometric data (S, d, V, r) is enough to further identify uniquely the geodesic cell complex C d and the collections of vertex and face circles. This is done via the weighted Delaunay cell decomposition construction. More precisely, given (i) a geometric surface (S, d), (ii) a finite set of points V ⊃ sing(d) on S and (iii) an assignment of disjoint vertex circle radii r : V → [0, ∞), one can uniquely generate (obtain) the corresponding r−weighted Delaunay cell complex C d , where each edge satisfies the local Delaunay property. In the process, the families of vertex and face circles naturally appear as part of the construction [6,21,19]. Alternatively, one can obtain the r−weighted Delaunay cell decomposition as the geodesic dual to the r-weighted Voronoi diagram, also known as the weighted power diagram with weights r [6]. A Voronoi cell in the case when d is Euclidean is defined as W d,r (i) = x ∈ S | d(x, i) 2 − r 2 i ≤ d(x, j) 2 − r 2 j for all j ∈ V . A Voronoi cell in the case when d is hyperbolic is defined as W d,r (i) = x ∈ S | cosh (r j ) cosh d(x, i) ≤ cosh (r i ) cosh d(x, j) for all j ∈ V . The circle pattern problem and the main resultLet us fix an arbitrary hyper-ideal circle pattern on S and let this pattern be determined by the data (S, d, V, r). Figure 2a depicts (a portion of) a hyper-ideal circle pattern. For each vertex i ∈ V one can define Θ i > 0 to be the cone angle of the cone-metric d at vertex i. Furthermore, since S is a closed surface, each edge ij ∈ E d is the common edge of exactly two faces from the r-weighted Delaunay cell-complex C d

Discrete Uniformization of Polyhedral Surfaces with Non-positive Curvature and Branched Covers over the Sphere via Hyper-ideal Circle Patterns

“…Following the terminology of [26] (see also [11]), one can define what Schlenker calls an admissible domain. Figure 4, denoted by the symbol Ω and shaded in grey.…”

“…As shown in [11], the angle data polytopes can be described via strict admissible domains instead of admissible domains. Simply, the admissible domains which are not strict do not add more restrictions to the angle data.…”

“…Theorem 4 guarantees the unique geometric realizability of the data (C, θ, Θ). In the following two sections 10 and 11 we describe the variational principle which plays a central role in the verification of Theorem 4 (see [11]) and provides a method for the construction of hyper-ideal circle patterns. All constructions and notations follow closely the exposition of [11].…”

“…First, we describe the space of (labelled) decorated triangles in H 2 considered up to hyperbolic isometries, with predetermined fixed splitting of the triangles' vertices V ∆ = {i, j, k} into V 1 ∆ and V 0 ∆ . Then, up to H 2 -isometry, a decorated triangle ∆ = ijk can be uniquely represented in three different ways [11]. The most natural way is to give the triangles edge lengths and vertex radii (l, r) ∆ = (l ij , l jk , l ki , r k , r i , r j ) ∈ ER ∆ , where the first three are positive and satisfy all three strict triangle inequalities l uv < l vw + l wu , as well as l uv > r u + r v for all u = v = w ∈ {i, j, k}, while the last three satisfy r u > 0 for u ∈ V 1 ∆ and r u = 0 for u ∈ V 0 ∆ .…”

“…The most natural way is to give the triangles edge lengths and vertex radii (l, r) ∆ = (l ij , l jk , l ki , r k , r i , r j ) ∈ ER ∆ , where the first three are positive and satisfy all three strict triangle inequalities l uv < l vw + l wu , as well as l uv > r u + r v for all u = v = w ∈ {i, j, k}, while the last three satisfy r u > 0 for u ∈ V 1 ∆ and r u = 0 for u ∈ V 0 ∆ . The second way of uniquely representing a decorated triangle is by its six angles (α ∆ , [26,31,11]. Here α ∆ uv is the angle between the geodesic edge uv of ∆ = ijk and its face circle c ∆ measured inside c ∆ and outside ∆, and β ∆ v is the interior angle of the triangle at its vertex v ∈ V ∆ (refer to Figure 1).…”