Euclidean Plane Research Articles

Pairs in discrete lattice orbits with applications to Veech surfaces (with an appendix by Jon Chaika)

Let \Lambda_{1} , \Lambda_{2} be two discrete orbits under the linear action of a lattice \Gamma<\mathrm{SL}_{2}(\mathbf{R}) on the Euclidean plane. We prove a Siegel–Veech-type integral formula for the averages \sum_{{\bf x}\in\Lambda_1}\sum_{{\bf y}\in\Lambda_2}f({\bf x},{\bf y}), from which we derive new results for the set S_{M} of holonomy vectors of saddle connections of a Veech surface M . This includes an effective count for generic Borel sets with respect to linear transformations, and upper bounds on the number of pairs in S_{M} with bounded determinant and on the number of pairs in S_{M} with bounded distance. This last estimate is used in the appendix to prove that for almost every (\theta,\psi)\in S^{1}\times S^{1} the translation flows F_{\theta}^{t} and F_{\psi}^{t} on any Veech surface M are disjoint.

Lie–Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications

Abstract We propose a generalization of two classes of Lie–Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie–Hamilton systems on Riemannian spaces (flat 2-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley–Klein spaces. These curved Lie–Hamilton classes allow us to generalize naturally the Riccati, Kummer–Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.

Symmetries of a General Polygon

Summary Most basic texts on group theory include a brief discussion of the symmetry group of a regular polygon in the Euclidean plane, but few go beyond this. Here we discuss the symmetry group of a general polygon by using ideas from geometry and number theory.

Approximation algorithms for minimum ply covering of points with unit squares and unit disks

Given a set P of points and a set U of geometric objects in the Euclidean plane, a minimum ply cover of P with U is a subset of U that covers P and minimizes the number of objects that share a common intersection, called the minimum ply cover number of P with U. Biedl et al. (2021) [9] showed that for both unit squares and unit disks, determining the minimum ply cover number for a set of points is NP-hard. They gave polynomial-time 2-approximation algorithms for the special case when the minimum ply cover number is constant, and asked whether there exists polynomial-time O(1)-approximation algorithms for these problems. In this paper, we settle the question posed by Biedl et al. by providing polynomial-time O(1)-approximation algorithms for the minimum ply cover problem for both unit squares and unit disks.

Analytical railway line capacity methods from graph-theoretical and polyhedral perspectives

ABSTRACT This paper rigorously addresses the consumed capacity underestimation caused by division-based methods in graph theory and polyhedral geometry. Graph-theoretically, such underestimation occurs because subgraphs in shorter sections approximate their combined graph. Our polyhedral study suggests that the capacity analysis is computationally ‘easy’ because its linear programme in the complete section has an orthogonal projection on the Euclidean plane containing five proper faces, one being the objective-defined one-dimensional optimal facet. Geometrically, we conclude that the polyhedra of the linear programmes in shorter and complete sections have parallel facets, with the former nested within the latter. Consequently, the optimal values of the former cannot exceed those of the latter. With these theoretical insights, we derive a train operation labelling algorithm that takes proven linear time. Beyond enabling faster capacity computation, the findings deepen our understanding of the connection between infrastructure capacity utilisation and timetable structure, ultimately leading to more efficient railway optimisation.

A Fourier approach to tomographic reconstruction of tensor fields in the plane

We consider the problem of inversion of the X-ray transform for sums of 1-forms and symmetric 2-tensor fields. Such a problem arises after linearization of a related travel time tomography problem, described via Mañé's action potential of the energy level 1/2 for a magnetic flow. In a strictly convex bounded domain in the Euclidean plane, we show when and how to recover simultaneously both unknown 1-tensor and symmetric 2-tensor field uniquely from measurement of radiating flux at the boundary. The approach to reconstruction is based on the Cauchy problem for a Beltrami-like equation associated with A-analytic maps.

From Tilings of Orientable Surfaces to Topological Interlocking Assemblies

A topological interlocking assembly (TIA) is an assembly of blocks together with a non-empty subset of blocks called the frame such that every non-empty set of blocks is kinematically constrained and can therefore not be removed from the assembly without causing intersections between blocks of the assembly. TIA provides a wide range of real-world applications, from modular construction in architectural design to potential solutions for sound insulation. Various methods to construct TIA have been proposed in the literature. In this paper, the approach of constructing TIA by applying the Escher trick to tilings of orientable surfaces is discussed. First, the strengths of this approach are highlighted for planar tilings, and the Escher trick is then exploited to construct a planar TIA that is based on the truncated square tiling, which is a semi-regular tiling of the Euclidean plane. Next, the Escher-Like approach is modified to construct TIAs that are based on arbitrary orientable surfaces. Finally, the capabilities of this modified construction method are demonstrated by constructing TIAs that are based on the unit sphere, the truncated icosahedron, and the deltoidal hexecontahedron.

Conformal Image Viewpoint Invariant

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R3,1 leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ∐αHα of R3,1 by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ∐αHα for the Minkowski metric of the ambient space.

Streaming Algorithms for Geometric Steiner Forest

We consider a generalization of the Steiner tree problem, the Steiner forest problem , in the Euclidean plane: the input is a multiset \(X\subseteq{\mathbb{R}}^{2}\) , partitioned into \(k\) color classes \(C_{1},\ldots,C_{k}\subseteq X\) . The goal is to find a minimum-cost Euclidean graph \(G\) such that every color class \(C_{i}\) is connected in \(G\) . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to \(X\) . Each input point \(x {\in} X\) arrives with its color \(\mathsf{color}(x) {\in} [k]\) , and as usual for dynamic geometric streams, the input is restricted to the discrete grid \(\{1,\ldots,\Delta\}^{2}\) . We design a single-pass streaming algorithm that uses \(\operatorname{poly}(k\cdot\log\Delta)\) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio \(\alpha_{2}\) (currently \(1.1547\leq\alpha_{2}\leq 1.214\) ). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when \(k=1\) , and it is a major open question to improve the ratio to \(1+\varepsilon\) even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires \(\Omega(k)\) bits of space.

Packing unequal disks in the Euclidean plane

A packing of disks in the plane is a set of disks with disjoint interiors. This paper is a survey of some open questions about such packings. It is organized into five themes: compacity, conjugacy, density, uniformity and computability.

Weighted means and an analytic characterization of discs

Weighted means are obtained for solutions of the two-dimensional Helmholtz and modified Helmholtz equations and also for harmonic functions. The presence of a logarithmic weight reduces the coefficient in the last two mean value identities. A new theorem characterizing disks in the Euclidean plane R 2 \mathbb {R}^2 analytically is proved; it is based on the weighted mean value property of solutions to the modified Helmholtz equation.

On uniform edge-n-colorings of tilings.

An edge-n-coloring of a uniform tiling {\cal T} is uniform if for any two vertices of {\cal T} there is a symmetry of {\cal T} that preserves the colors of the edges and maps one vertex onto the other. This paper gives a method based on group theory and color symmetry theory to arrive at uniform edge-n-colorings of uniform tilings. The method is applied to give a complete enumeration of uniform edge-n-colorings of the uniform tilings of the Euclidean plane, for which the results point to a total of 114 colorings, n = 1, 2, 3, 4, 5. Examples of uniform edge-n-colorings of tilings in the hyperbolic plane and two-dimensional sphere are also presented.

Morphometry on the sphere: Cartesian and irreducible Minkowski tensors explained and implemented

Minkowski tensors are comprehensive shape descriptors that robustly capture n-point information in complex random geometries and that have already been extensively applied in the Euclidean plane. Here, we devise a framework for Minkowski tensors on the sphere. We first advance the theory by introducing irreducible Minkowski tensors, which avoid the redundancies of previous representations. We, moreover, generalize Minkowski sky maps to the sphere. These maps are a concept of local anisotropy, which easily adjusts to masked data. We demonstrate the power of our new procedure by applying it to simulations and real data of the Cosmic Microwave Background, finding an anomalous region close to the well-known Cold Spot. The accompanying open-source software, litchi, used to generate these maps from data in the HEALPix-format is made publicly available to facilitate broader integration of Minkowski maps in other fields, such as fluid demixing, porous structures, or geosciences more generally.

Mutual visibility of luminous robots despite angular inaccuracy

We initiate the study of the Mutual Visibility problem using N opaque luminous point robots that have inaccurate movements. Each robot operates in Look-Compute-Move cycles and has a persistent light attached to it to have a weak form of communication between robots using a constant number of colors. The inaccuracy for a robot r is an angular deviation from its target point T to a point T′ such that the angle ∠TrT′<90∘. The problem becomes unsolvable if this angle is ≥90∘. From any initial configuration of the robots on the Euclidean plane, the problem aims to arrange the robots in a configuration such that any two robots are visible to each other. We assume that the robots agree on one coordinate axis. We present two collision-free algorithms, a 2 color algorithm (which is optimal in the number of colors used) for semi-synchronous setting and a 3 color algorithm for asynchronous setting, both of which run in O(N) epochs. We also study the problem in the presence of mobile faulty robots. A robot can exhibit both mobility failure and angular inaccuracies in its movement. We present a fault-tolerant algorithm that aims to bring the robots in a configuration where no three non-faulty robots are collinear, and no faulty robot lies between two non-faulty robots. This algorithm uses 10 colors and takes O(N) epochs under asynchronous settings.

On the terminal conditions of the two cutters and a fugitive ship differential game with non-zero capture radius and different players’ speed ratio

We study a pursuit-evasion differential game in which three agents move with simple motion on the Euclidean plane. Two of them, the cutters (pursuers), aim to capture a fugitive ship (the evader) as soon as possible. Having an opposite goal, the fugitive ship seeks to avoid capture for as long as possible. The game ends when the distance between the fugitive and at least one of the cutters is smaller than a given value. We have divided the game into two cases: case 1, when all players have the same speed, and case 2, when the evader is faster than the pursuers. Unlike previous work, our main innovations are as follows. For case 1, we present a solution obtained using exclusively differential game techniques. The game of kind is solved by establishing a study of the barrier that defines the winner of the game. In addition, we obtained time-optimal strategies for all players, proving that they do not switch controls. For case 2, we obtain the primary solution and exhibit an example showing the existence of control switches.

On Vertices of Frontals in the Euclidean Plane

On Vertices of Frontals in the Euclidean Plane

Z2 classification of FTR symmetric differential operators and obstruction to Anderson localization

This paper concerns the Z2 classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.

Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing

Given a random map [Formula: see text], we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards.

Scalable codes with locality and availability derived from tessellation via [7, 3, 4] Simplex code graph

Abstract A new family of scalable codes with locality and availability for information repair in data storage systems for e-health applications was presented recently. The construction was based on a graph of the [7, 3, 4] Simplex code. In this paper it is shown that the construction can be generalized via tessellation in a Euclidian plane. The codes obtained have new interesting recoverability properties. They can in some cases repair damage to many storage nodes in multiple connected graphs via sequential decoding, which is similar to healing wounds in biological systems. The advantages of the original codes, namely the availability, functionality, efficiency and high data accessibility, will be preserved also in these new codes. The computational complexity and communication costs of their incrementation will remain constant and modest. These codes could be adapted to disaster recovery because it is straightforward to place the nodes so that the graph is easily mapped on a real structure in space.

Inversion of the attenuated momenta ray transform of planar symmetric tensors

We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim’s A-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of one-tensors.