A rigidity theory is developed for countably infinite simple graphs in \({\mathbb {R}}^d\). Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in \({\mathbb {R}}^2\) and Tay’s multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in \({\mathbb {R}}^d\). Analogous...

Zigzags in thin chamber complexes are investigated, in particular, all zigzags in the Coxeter complexes are described. Using this description, we show that the lengths of all zigzags in the simplex \(\alpha _{n}\), the cross-polytope \(\beta _{n}\), the 24-cell, the icosahedron and the 600-cell are equal to the Coxeter numbers of \(\mathsf{A}_{n}\), \(\mathsf{B}_{n}=\mathsf{C}_{n...

Classical methods to model topological properties of point clouds, such as the Vietoris–Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in \(\mathbb {R}^d\), we obtain a O(d)-approximation whose k-skeleton has size...

Quasi-triangulations of a non-orientable surface were introduced by Dupont and Palesi (J Algebr Comb 42(2):429–472, 2015). The quasi-arc complex provides an intricate description of the combinatorics of these quasi-triangulations. This is the simplicial complex where vertices correspond to quasi-arcs and maximal simplices to quasi-triangulations. We prove that when the quasi-arc...

New classes of infinite bond-node structures are introduced, namely string-node nets and meshes, a mesh being a string-node net for which the nodes are dense in the strings. Various construction schemes are given including the minimal extension of a (countable) line segment net by a countable scaling group. A linear mesh has strings that are straight lines and nodes given by the...

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the Gauss...

An embedding \(i \mapsto p_i\in \mathbb {R}^d\) of the vertices of a graph G is called universally completable if the following holds: For any other embedding \(i\mapsto q_i~\in \mathbb {R}^{k}\) satisfying \(q_i^{T}q_j = p_i^{T}p_j\) for \(i = j\) and i adjacent to j, there exists an isometry mapping \(q_i\) to \(p_i\) for all \( i\in V(G)\). The notion of universal...

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least \(n^2/4 - O(n)\) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this...

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding \(\frac{1}{2}\). We conclude that all regular maps of this type belong to a family of maps...

In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii \(r_1\), \(\ldots \), \(r_n\) in the plane, it is always possible to cover them by a disk of radius \(R = \sum r_i\), provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially...

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original \(O(n^2 \log n)\) algorithm by Alt and Godau for computing the Fréchet...

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.

Let S be a subset of \(\mathbb {R}^d\) with finite positive Lebesgue measure. The Beer index of convexity \({\text {b}}(S)\) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio \({\text {c}}(S)\) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We...

We give a concise definition of mitered offset surfaces for nonconvex polytopes in \({\mathbbm {R}}^3\), along with a proof of existence and a discussion of basic properties. These results imply the existence of 3D straight skeletons for general nonconvex polytopes. The geometric, topological, and algorithmic features of such skeletons are investigated, including a classification...

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet...

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map \(f:K\rightarrow {\mathbb {R}}^n\) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within \(L_\infty \) distance r from f for a given \(r>0\). The main drawback of the approach is...

We investigate in this paper the relation between Apollonian d-ball packings and stacked \((d+1)\)-polytopes for dimension \(d\ge 3\). For \(d=3\), the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there is no six 4-cliques sharing a 3-clique. For higher dimension, we have...

We give various quantitative versions of Łojasiewicz inequalities for semialgebraic sets and mappings, both in the local and global case.

We present results referring to the Hadwiger–Nelson problem which asks for the minimum number of colors needed to color the plane with no two points at distance 1 having the same color. Exoo considered a more general problem concerning graphs \(G_{[a,b]}\) with \(\mathbb {R}^2\) as the vertex set and two vertices adjacent if their distance is in the interval [a, b]. Exoo...

Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of \(\mathbb {D}^2\times \mathbb {R}^n\) equals \(\pi /\sqrt{12}\) for all natural numbers n.

We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject...

We characterise finite and infinitesimal rigidity for bar-joint frameworks in \({\mathbb {R}}^d\) with respect to polyhedral norms (i.e. norms with closed unit ball \({\mathcal {P}}\), a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in \({\mathbb {R}}^d\) which are well-positioned with respect to \({\mathcal...