Euclidean Plane Research Articles

Remarks on a stochastic geometric model for interference-limited cellular communications

A plethora of stochastic geometry (SG) models have been developed for cellular communications, especially in the context of internet of things (IoT) applica tions. A typical assumption in such models is that base stations (BS) are de ployed in the Euclidean plane as a spatial poisson point process (PPP) of some density λ, with each communicating equipment transmitting at some power p. The usual objective of these models is to characterize the cellular coverage prob ability in both the downlink (DL) and uplink (UL) directions. In this article we expose, in the form of four remarks, the peculiar behavior of a baseline stochas tic geometric model of an interference-limited cellular system. Specifically, we reveal that under some assumptions, the coverage probability in both the UL and DL directions for this system is independent of both λ and p, flagrantly contra dicting intuition. The aim of the article is by no means to invalidate the use of SG in modeling communications systems, but rather to point out that such modeling may not be adequate all the time.

Chen-Chvátal’s conjecture for graphs with restricted girth

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

A Mazurkiewicz set containing the graph of a Sierpiński–Zygmund function

Abstract It is shown that there exists a Mazurkiewicz subset of the Euclidean plane containing the graph of some Sierpiński–Zygmund function.

On the power of bounded asynchrony: convergence by autonomous robots with limited visibility

AbstractA distributed algorithm $${\mathcal {A}}$$ A solves the Point Convergence task if an arbitrarily large collection of entities, starting in an arbitrary configuration, move under the control of $${\mathcal {A}}$$ A to eventually form and thereafter maintain configurations in which the separation between all entities is arbitrarily small. This fundamental task in the standard $$\mathcal {OBLOT}$$ OBLOT model of autonomous mobile entities has been previously studied in a variety of settings, including full visibility, exact measurements (including distances and angles), and synchronous activation of entities. Our study concerns the minimal assumptions under which entities, moving asynchronously with limited and unknown visibility range and subject to limited imprecision in measurements, can be guaranteed to converge in this way. We present an algorithm operating under these constraints that solves Point Convergence, for entities moving in two or three dimensional space, with any bounded degree of asynchrony. We also prove that under similar realistic constraints, but unbounded asynchrony, Point Convergence in the plane is not possible in general, contingent on the natural assumption that algorithms maintain the (visible) connectivity among entities present in the initial configuration. This variant, that we call Cohesive Convergence, serves to distinguish the power of bounded and unbounded asynchrony in the control of autonomous mobile entities, settling a long-standing question whether in the Euclidean plane synchronously scheduled entities are more powerful than asynchronously scheduled entities.

Models of Klein Surface Obstruction Graphs

The task of researching the structure of graphs of given connectivity, which are obstructions for a given surface of non-oriented kind, and building their models, from which obstruction graphs are formed by removing or compressing a set of edges, is considered. The issue of edge coverage of an obstruction graph of a given kind with a minimum number of quasi-stars with centers – planar graphs that have given sets of points and all edges are significant with respect to the reachability number 2 on the Euclidean plane and has reachability on the projective plane or Klein surface, is considered. K4, K2,3 or a degenerate graph. The task of researching the structure of graphs of undirected kind was considered [4–6]. In [7], the set of minors for the projective plane was compressed to 12 basic minors using the method of relative components, and a set of 62 minors of the Klein surface was constructed. To do this, we considered all non-isomorphic minimal embeddings of each of the basic minors and found the set of all different pairs of vertices that are reachable on the projective plane during the operations of removing or compressing an arbitrary edge of this graph to a point, then a pair of non-adjacent graph vertices was attached to the selected pair of points. In [8], the number of 2-connected obstruction graphs for the Klein surface was calculated, part of the diagrams of these graphs is given in [10]. Note that the following definition of the cell distance is similar to that in [11].Our approach, as a continuation of [9], will consist in finding the edge covering of an obstruction graph of a given kind by the minimum number of subgraphs of the covering from the number of quasi-stars with centers - graphs with essential edges relative to the number of reachability or nonorientable genus during compression to a point or removal operations edges relative to a given set of points with reachability number 2 relative to the Euclidean plane and reachable on projective planes or Klein surfaces, for example, these are subsets of the set of points of graphs K4, K2,3, K5\e, Kr, r >= 2, or graph-obstructions of the projective plane. We also found the necessary conditions for constructing obstruction graphs for the Klein surface by identifying pairs of center points and hanging vertices of three quasi-stars, thus we have the basis of an algorithm for constructing a larger number of obstruction graphs for the Klein surface. Hypothetically, a graph-obstruction of a given nonorientable genus has the form of a cylindrical surface with n, n >= 2, disks-bases and a side part, which can have common sets of points on the boundaries and on which are embedded, at least in part, the graph-centers of quasi-stars having a given set of reachability points 2 on the Euclidean plane, and on the side surface there are hanging edges that intersect on the plane and are inserted without crossing with the help of Mobius strips glued to the side surface. At the same time, the edges will have at least two nesting options in the side part of the cylindrical surface, but no more than the number of glued Mobius strips, thanks to which each hanging edge will nest on the Mobius strip, either with only one edge or with two adjacent edges. We have found the necessary conditions for constructing models of obstruction graphs for the Klein surface by identifying pairs of centers and hanging vertices of three quasi-stars, thus we have the basis of an algorithm for constructing a larger number of obstruction graphs for the Klein surface. The main result: statements 1, 2, 3 and the algorithm for constructing models of 3-connected graph-obstructions of the Klein surface. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.

On Uniqueness in Steiner Problem

Abstract We prove that the set of $n$-point configurations for which the solution to the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff dimension of the set of $n$-point configurations for which at least two locally minimal trees have the same length is also at most $2n-1$. The methods we use essentially rely upon the theory of subanalytic sets developed in [ 1]. Motivated by this approach, we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replaced by an arbitrary analytic Riemannian manifold $M$. In this setup, we argue that the set of configurations possessing two locally-minimal trees of the same length either has dimension equal to $n \dim M - 1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to the above-mentioned results, we study the set of $n$-point configurations for which there is a unique solution to the Steiner problem in $\mathbb{R}^{d}$. We show that this set is path-connected.

Singular matrices and pairwise-tangent circles

The idea of using the generalised inverse of a singular matrix A to solve the matrix equation Ax = b has been discussed in the earlier papers [1, 2, 3, 4] in the Gazette. Here we discuss three simple geometric questions which are of interest in their own right, and which illustrate the use of the generalised inverse of a matrix. The three questions are about polygons and circles in the Euclidean plane. We need not assume that a polygon is a simple closed curve, nor that it is convex: indeed, abstractly, a polygon is just a finite sequence (v1, …, vn) of its distinct, consecutive, vertices. It is convenient to let vn + 1 = v1 and (later) Cn + 1 = C1.

TSCAPE: time series clustering with curve analysis and projection on an Euclidean space

The ever-growing use of digital systems has led to the accumulation of vast datasets, particularly time series, depicting the temporal evolution of variables and systems. Analysing these time series presents a tremendous challenge due to their inherent complexity and heterogeneity. Addressing an industrial need in the pharmaceutical wholesale sector, this paper introduces a new clustering method for time series: TSCAPE. The TSCAPE method uses a distance matrix calculated using dynamic time warping, followed by multidimensional scaling to project time series into a 2D Euclidean space, thus improving the last clustering stage by K-Means. Unlike conventional techniques, this approach, based on clustering of representation of distances in a Euclidean plane rather than on the curve shape, directly enhances the efficiency of the clustering process. The methodology exhibits significant potential for diverse applications, accommodating varied data types and irregular time series shapes. The research compares multiple variants and proposes metrics to assess their effectiveness on two open-access datasets. The results demonstrate the method's superiority over “only distance comparison clustering techniques”, like dynamic time warping and K-Means, with future prospects aimed at predictive applications and refining the clustering process by exploring alternative, more powerful clustering algorithms.

Neural networks with functional inputs for multi-class supervised classification of replicated point patterns

AbstractA spatial point pattern is a collection of points observed in a bounded region of the Euclidean plane or space. With the dynamic development of modern imaging methods, large datasets of point patterns are available representing for example sub-cellular location patterns for human proteins or large forest populations. The main goal of this paper is to show the possibility of solving the supervised multi-class classification task for this particular type of complex data via functional neural networks. To predict the class membership for a newly observed point pattern, we compute an empirical estimate of a selected functional characteristic. Then, we consider such estimated function to be a functional variable entering the network. In a simulation study, we show that the neural network approach outperforms the kernel regression classifier that we consider a benchmark method in the point pattern setting. We also analyse a real dataset of point patterns of intramembranous particles and illustrate the practical applicability of the proposed method.

Extremizers and Stability of the Betke–Weil Inequality

Let K be a compact convex domain in the Euclidean plane. The mixed area A(K,−K) of K and −K can be bounded from above by 1/(6 3)L(K)2, where L(K) is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil [5]. They also showed that if K is a polygon, then equality holds if and only if K is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality 6 3A(K,−K)≤L(K)2.

Solving olympiad geometry without human demonstrations

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1–4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

Noncommutativity in Configuration Space Induced by a Conjugate Magnetic Field in Phase Space

An external magnetic field in configuration space coupled to quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite ∗-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore, when the positive definite ∗-product is adapted to the conjugate magnetic field, the coordinate operators span a Fock algebra of which the coherent states are the analogues of the structureless points in a commutative configuration space. These results generalise and justify a posteriori within ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.

On Directed Length Ratios in the Lorentz-Minkowski Plane

The linear structure of the Lorentz-Minkowski plane is almost the same as Euclidean plane. But, there is one different aspect. These planes have different distance functions. So, it can be interesting to study the Lorentz analogues of topics that include the distance concept in the Euclidean plane. Thus, in this study, we show that the relationship between Euclidean and Lorentz distances is given depending on the slope of the line segment. Following, we investigate Lorentz analogues of Thales’ theorem, Angle Bisector theorems, Menelaus’ theorem and Ceva’s theorem.

Stability of minimal surfaces in the sub-Riemannian manifold $\widetilde{E(2)}$

In the paper we study smooth oriented surfaces in the universal covering space of the group of orientation-preserving Euclidean plane isometries, which has a three-dimensional sub-Riemannian manifold structure. This structure is constructed as a restriction of the Euclidean metric on the group to some completely non-integrable left invariant distribution. The sub-Riemannian area of a surface is then defined as the integral of the length of its unit normal field projected orthogonally onto this distribution. We calculate the first variation formula of the sub-Riemannian surface area and derive the minimality criterion from it. Here we call a surface minimal if it is a critical point of the sub-Riemannian area functional under normal variations with compact support. We show that the minimality in this case is not equivalent to the vanishing of the sub-Riemannian mean curvature. We then prove that a Euclidean plane is minimal if and only if it is parallel or orthogonal to the $z$-axis (where the $z$-coordinate corresponds to the rotation angle of an isometry). Also we obtain the minimality condition for a graph and give examples of minimal graphs. The examples considered in the paper demonstrate, in particular, that the minimality of a surface in the Riemannian (in this case Euclidean) sense does not imply its sub-Riemannian minimality, and vice versa. Next, we consider the stability of minimal surfaces. For this purpose, we derive the second variation formula of the sub-Riemannian area and show with it that minimal Euclidean planes are stable. We introduce a class of surfaces for which the tangent planes are perpendicular to the planes of the sub-Riemannian structure, and call them vertical surfaces. In particular, for such surfaces the second variation formula is simplified significantly. Then we prove that complete connected vertical minimal surfaces are either Euclidean planes or helicoids and that helicoids are unstable. This implies a following Bernstein type result: a complete connected vertical minimal surface is stable if and only if it is a Euclidean plane orthogonal to the $z$-axis.

When is a Planar Rod Configuration Infinitesimally Rigid?

AbstractWe investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

The adjoint map of Euclidean plane curves and curvature problems

The adjoint map of a pair of naturally parametrized curves in the Euclidean plane is studied from the point of view of the curvature. A main interest is when the given curve and its adjoint curve share the same natural parameter and the same curvature. For the general linear second order differential equation we introduce a function expressing the deformation of curvatures induced by the adjoint map.

Some New Properties of Cyclotomic Cliques Arrangements

In this paper, we study the arrangement of lines in the euclidean plane constructed from the geometric clique graph generated by the regular -gon. The vertices of this clique arrangement are located on finite concentric circles that we call orbits. We focus especially on the number of orbits and the number of vertices inside and outside the regular -gon. Combinatorics in finite sets of quadruplets of integers provide information on the way the orbits are distributed. Next, using cyclotomic fields, we give galoisian properties of the radii of the orbits and their cardinalities. Keywords: arrangement of lines in the plane, cyclotomic fields, geometric graphs, Galois theory

Some extremal problems for polygons in the Euclidean plane

Some extremal problems for polygons in the Euclidean plane

An efficient 3-approximation algorithm for the Steiner tree problem with the minimum number of Steiner points and bounded edge length.

We present improved algorithms for the Steiner tree problem with the minimum number of Steiner points and bounded edge length. Given n terminal points in a 2D Euclidean plane and an edge length bound, the problem asks to construct a spanning tree of n terminal points with minimal Steiner points such that every edge length of the spanning tree is within the given bound. This problem is known to be NP-hard and has practical applications such as relay node placements in wireless networks, wavelength-division multiplexing(WDM) optimal network design, and VLSI design. The best-known deterministic approximation algorithm has O(n3) running time with an approximation ratio of 3. This paper proposes an efficient approximation algorithm using the Voronoi diagram that guarantees an approximation ratio of 3 in O(n log n) time. We also present the first exact algorithm to find an optimal Steiner tree for given three terminal points in constant time. Using this exact algorithm, we improve the 3-approximation algorithm with better performance regarding the number of required Steiner points in O(n log n) time.

Enumerating combinatorial resultant trees

A 2D rigidity circuit is a minimal graph G=(V,E) supporting a non-trivial stress in any generic placement of its vertices in the Euclidean plane. All 2D rigidity circuits can be constructed from K4 graphs using combinatorial resultant (CR) operations. A combinatorial resultant tree (CR-tree) is a rooted binary tree capturing the structure of such a construction.The CR operation has a specific algebraic interpretation, where an essentially unique circuit polynomial is associated to each circuit graph. Performing Sylvester resultant operations on these polynomials is in one-to-one correspondence with CR operations on circuit graphs. This mixed combinatorial/algebraic approach led recently to an effective algorithm for computing circuit polynomials. Its complexity analysis remains an open problem, but it is known to be influenced by the depth and shape of CR-trees in ways that have only partially been investigated.In this paper, we present an effective algorithm for enumerating all the CR-trees of a given circuit with n vertices. Our algorithm has been fully implemented in Mathematica and allows for computational experimentation with various optimality criteria in the resulting, potentially exponentially large collections of CR-trees.