Euclidean Plane Research Articles

Maximum Spanning Trees in Normed Planes

Extending some properties from the Euclidean plane to any normed plane, we show the validity of the Monma–Paterson–Suri–Yao algorithm for finding the maximum-weighted spanning tree of a set of n points, where the weight of an edge is the distance between the end points measured by the norm and there are not repeated distances. For strictly convex normed planes, we expose a strategy for moving slightly the points of the set in order to obtain distinct distances.

Rigidity of Riemannian embeddings of discrete metric spaces

Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We prove that if the distances in $\mathcal{S}$ correspond to the distances in a $2$-dimensional lattice, or more generally in an arbitrary net in $\mathbb{R}^2$, then $M$ is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of $\mathbb{Z}^3$ that strictly contains $\mathbb{Z}^2 \times \{ 0 \}$ cannot be isometrically embedded in any complete Riemannian surface.

Memory-Adjustable Navigation Piles with Applications to Sorting and Convex Hulls

We consider space-bounded computations on a random-access machine, where the input is given on a read-only random-access medium, the output is to be produced to a write-only sequential-access medium, and the available workspace allows random reads and writes but is of limited capacity. The length of the input is N elements, the length of the output is limited by the computation, and the capacity of the workspace is O ( S ) bits for some predetermined parameter S ≥ lg N . We present a state-of-the-art priority queue—called an adjustable navigation pile —for this restricted model. This priority queue supports M inimum in O (1) time, C onstruct in O ( N ) time, and E xtract - min in O ( N / S + lg S ) time for any S ≥ lg N . The priority queue can be further augmented in O ( N ) time to deal with a batch of at most S elements in a specified range of values at a time, and allow to I nsert (activate) or E xtract (deactivate) an element among these elements, such that I nsert and E xtract take O ( N / S + lg S ) time for any S ≥ lg N . We show how to use our data structure to sort N elements and to compute the convex hull of N points in the Euclidean plane in O ( N 2 / S + N lg S ) time for any S ≥ lg N . Following a known lower bound for the space-time product of any branching program for finding unique elements, both our sorting and convex-hull algorithms are optimal. The adjustable navigation pile has turned out to be useful when designing other space-efficient algorithms, and we expect that it will find its way to yet other applications.

On the effectiveness of the genetic paradigm for polygonization

A polygon is simple if it is a closed chain of straight line segments that do not self-intersect. Given a finite set P of input points in the Euclidean plane, the search for a simple polygon with vertex set P is a very well-known and studied computational problem, referred to as the simple polygonization. Its optimization version, that requires to find a simple polygon having either minimum or maximum area, is known to be NP-hard. Moreover, no bounded approximation algorithm is known for the minimization flavor, while the maximization one admits an algorithm guaranteeing 12 worst-case approximation ratio. In this work, we design a new algorithm to practically attack both the optimization problems, based on the genetic paradigm. We demonstrate its effectiveness through an extensive experimental evaluation that employs a reference test-bed set of input instances.

Incongruent equipartitions of the plane into quadrangles of equal perimeters

Motivated by a question of R. Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the plane by mutually incongruent triangles of unit area that are arbitrarily close to the periodic vertex-to-vertex tiling by equilateral triangles.

Constant Angle Surfaces in the Lorentzian Warped Product Manifold $$-I \times _{f} \mathbb {E}^2$$

In this work, we study constant angle space-like and time-like surfaces in the 3-dimensional Lorentzian warped product manifold $$-I \times _{f} \mathbb {E}^2$$ with the metric $${\tilde{g}} = - \mathrm{d}t^2 + f^2(t) (\mathrm{d}x^2 + \mathrm{d}y^2)$$ , where I is an open interval, f is a strictly positive function on I, and $$\mathbb {E}^2$$ is the Euclidean plane. We obtain a classification of all constant angle space-like and time-like surfaces in $$-I \times _{f} \mathbb {E}^2$$ . In this classification, we determine space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we obtain some results on constant angle space-like and time-like surfaces of the de Sitter space $$\mathbb {S}^3_1 (1)$$ .

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Inspired by the seminal works of Khuller et al. (SIAM J. Comput. 25(2), 355–368 (1996)) and Chan (Discrete Comput. Geom. 32(2), 177–194 (2004)) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-K spanning tree is a degree-K spanning tree whose largest edge-length is minimum. Let $$\beta _K$$ be the supremum ratio of the largest edge-length of the bottleneck degree-K spanning tree to the largest edge-length of the bottleneck spanning tree, over all finite point sets in the Euclidean plane. It is known that $$\beta _5=1$$ , and it is easy to verify that $$\beta _2\geqslant 2$$ , $$\beta _3\geqslant \sqrt{2}$$ , and $$\beta _4>1.175$$ . It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that $$\beta _2\leqslant 3$$ . The degree-3 spanning tree algorithm of Ravi et al. (25th Annual ACM Symposium on Theory of Computing, pp. 438–447. ACM, New York (1993)) implies that $$\beta _3\leqslant 2$$ . Andersen and Ras (Networks 68(4), 302–314 (2016)) showed that $$\beta _4\leqslant \sqrt{3}$$ . We present the following improved bounds: $$\beta _2\geqslant \sqrt{7}$$ , $$\beta _3\leqslant \sqrt{3}$$ , and $$\beta _4\leqslant \sqrt{2}$$ . As a result, we obtain better approximation algorithms for Euclidean bottleneck degree-3 and degree-4 spanning trees. As parts of our proofs of these bounds we present some structural properties of the Euclidean minimum spanning tree which are of independent interest.

Exotic Steiner chains in Miquelian Möbius planes of odd order

Abstract In the Euclidean plane, two circles that intersect or are tangent clearly do not carry a finite Steiner chain of circles. We show that such exotic Steiner chains exist in finite Miquelian Möbius planes of odd order. We obtain explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles for the existence, length, and number of Steiner chains.

Extremal Problems for Convex Curves with a Given Self Chebyshev Radius

The paper is devoted to some extremal problems for convex curves and polygons in the Euclidean plane referring to the relative Chebyshev radius. In particular, we determine the relative Chebyshev radius for an arbitrary triangle. Moreover, we derive the maximal possible perimeter for convex curves and convex n-gons of a given relative Chebyshev radius.

Perimeter Approximation of Convex Discs in the Hyperbolic Plane and on the Sphere

Eggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.

The minimum area spanning tree problem: Formulations, Benders decomposition and branch-and-cut algorithms

The Minimum Area Spanning Tree Problem (MASTP) is defined in terms of a complete undirected graph G, where every vertex represents a point in the two dimensional Euclidean plane. Associated to each edge, there is a disk placed right at its midpoint, with diameter matching the length of the edge. MASTP seeks a spanning tree of G that minimizes the area in the union of the disks associated to its edges. This paper presents Integer Programming (IP) approaches for MASTP, introduces pre-processing techniques to reduce the size of the formulations, and characterizes valid inequalities for reinforcing its Linear Programming Relaxation bounds. Several Branch-and-cut (BC) algorithms exploiting such ideas are introduced. Additionally, we also apply Benders Decomposition to one of these formulations. An accompanying BC method, that separates Benders optimality cuts, is also introduced. Aiming to save linear programming re-optimization times, that algorithm makes use of an early branching strategy. Given the set of valid inequalities used in the polyhedral representation of the problem and the best available upper bound for the optimal cost, it detects if the node cannot be pruned by bounds and then stops the cutting plane generation, in order to branch. Our algorithms manage to solve instances with up to only 15 vertices, suggesting thus that MASTP is hard to solve in practice, at least with the currently available methods. Thanks to the early branching strategy, the Benders based BC method obtained the best computational results, by far. It solved more instances to proven optimality than the other algorithms (31 out of 45 cases) and it is about three times faster than the second best performing algorithm.

Classification of Evolutoids and Pedaloids in Minkowski Space-time Plane

In this paper, we study the families of relatives of pedals and evolutes in the Minkowski spacetime plane R 2 1 . We obtain some relationships between these families which turn out to be different from Euclidean plane. Also, we classify and generalize these notions to the category of frontal curves in R 2 1 . Finally, some computational examples in support of our main results are given and plotted.

Some geometric characterizations of Pythagorean and Euclidean fields

In this paper some new geometric characterizations are given of the following algebraic properties of the underlying field K of an Euclidean plane $$E: -1$$ is a square in K, the group of squares is of index 2 in $$K^*$$ , K is Pythagorean, and K is Euclidean. In the first part ordinary incidence conditions of lines and circles in E are used for this purpose, while in the second these properties are characterized by comparing four betweenness relations introduced in E in a natural way.

Magnetic WKB constructions on surfaces

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially symmetric magnetic field, the eigenfunctions are approximated in an exponentially weighted space.

Affine self-similar solutions of the affine curve shortening flow I: The degenerate case

In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give descriptions of the solutions for the degenerate case. Some new special solutions for the affine curve shortening flow are found.

Higher order Wirtinger-type inequalities and sharp bounds for the isoperimetric deficit

Using Fourier analysis, we derive Wirtinger-type inequalities of arbitrary high order. As applications, we prove various sharp geometric inequalities for closed curves on the Euclidean plane. In particular, we obtain both sharp lower and upper bounds for the isoperimetric deficit.

Eigenfunctions and the integrated density of states on Archimedean tilings

We study existence and absence of \ell^2 -eigenfunctions of the combinatorial Laplacian on the 11 Archimedean tilings of the Euclidean plane by regular convex polygons. We show that exactly two of these tilings (namely the (3.6)^2 “kagome” tiling and the (3.12^2) tiling) have \ell^2 -eigenfunctions. These eigenfunctions are infinitely degenerate and are constituted of explicitly described eigenfunctions which are supported on a finite number of vertices of the underlying graph (namely the hexagons and 12-gons in the tilings, respectively). Furthermore, we provide an explicit expression for the Integrated Density of States (IDS) of the Laplacian on Archimedean tilings in terms of eigenvalues of Floquet matrices and deduce integral formulas for the IDS of the Laplacian on the (4^4) , (3^6) , (6^3) , (3.6)^2 , and (3.12^2) tilings. Our method of proof can be applied to other \mathbb Z^d -periodic graphs as well.

On the Analysis of Spatially Constrained Power of Two Choice Policies

We consider a class of power of two choice based assignment policies for allocating users to servers, where both users and servers are located on a two-dimensional Euclidean plane. In this framework, we investigate the inherent tradeoff between the communication cost, and load balancing performance of different allocation policies. To this end, we first design and evaluate a Spatial Power of two (sPOT) policy in which each user is allocated to the least loaded server among its two geographically nearest servers sequentially. When servers are placed on a two-dimensional square grid, sPOT maps to the classical Power of two (POT) policy on the Delaunay graph associated with the Voronoi tessellation of the set of servers. We show that the associated Delaunay graph is 4-regular and provide expressions for asymptotic maximum load using results from the literature. For uniform placement of servers, we map sPOT to a classical balls and bins allocation policy with bins corresponding to the Voronoi regions associated with the second order Voronoi diagram of the set of servers. We provide expressions for the lower bound on the asymptotic expected maximum load on the servers and prove that sPOT does not achieve POT load balancing benefits. However, experimental results suggest the efficacy of sPOT with respect to expected communication cost. Finally, we propose two non-uniform server sampling based POT policies that achieve the best of both the performance metrics. Experimental results validate the effectiveness of our proposed policies.

Окружности на комплексной плоскости

The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

FRACTAL REP TILES OF ℝ2 AND ℝ3 USING INTEGER MATRICES

This paper is devoted to the study of self-similar fractal rep tiles of the Euclidean planes [Formula: see text] and [Formula: see text] using integer matrices. We construct many examples appearing in the literature, as well as new examples by integer matrices. Several exotic and radially symmetric self-similar rep tiles and their variations are obtained with change of bases extending the work by Bandt and others. We also present a few examples of polyhedral self-similar rep tiles including variations.