Minkowski Plane Research Articles

Revisiting the foundations of Barbilianʼs metrization procedure

In the present work we prove that one of Barbilianʼs theorems from 1960 regarding the metrization procedure in the plane admits a natural extension depending on a bilinear form and the relative position of two Apollonian hyperspheres. This result allows us to pursue two fundamental ideas. First, that all the distances with constant curvature can be described by Barbilianʼs metrization principle. Secondly, that all the Riemannian metric corresponding to these distances can be obtained with the same unique procedure derived from the main theorem in the text (Theorem 2.5). We show how the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in the same way using Theorem 2.5, which appears here for the first time and is an extension of a Barbilian classical result (Barbilian, 1960 [7]). Furthermore, we obtain metrics corresponding to quadratic forms with signature that includes minus. By considering the norms provided by either Lorentz or Minkowski (pseudo-)inner product as influence functions, two oscillant distances can be generated in some subsets of Lorentz or Minkowski plane. The extension of 1960 Barbilianʼs theorem mentioned above allow us to obtain the metrics attached to these two Barbilian distances on corresponding subsets of Lorentz and Minkowski 2-dimensional spaces. The geometric study concludes that these metrics are generalized Lagrange metrics. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance is also proved.

On angular measures in Minkowski planes

We introduce the notion of angular measure in a Minkowski plane and prove that the plane has to be Euclidean if any two James-orthogonal vectors form an angle of value $${\frac{\pi}2}$$ , but need not be Euclidean if any two Birkhoff orthogonal vectors do so.

Asymptotics of generalized Hadwiger numbers

We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval B which have a common point with a 2-dimensional domain F having rectifiable boundary, extending previous work of L. Fejes Tóth, K. Böröckzy Jr., D. G. Larman, S. Sezgin, C. Zong and the authors. The asymptotics compute the length of the boundary ∂F in the Minkowski metric determined by B. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erdős, Oleksiv and Pesin for the Euclidean space to the Minkowski space.

Ellipses numbers and geometric measure representations

Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A(r) = π(a,b)r2, the number π(a,b) = abπ reflects a generalized circumference property U(a,b)(r) = 2π(a,b)r of the ellipses E(a,b)(r) with main axes of lengths 2ra and 2rb, respectively. In this sense, the number π(a,b) is an ellipse number w.r.t. the Minkowski functional r of the reference set E(a,b)(1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.

Approximating minimum Steiner point trees in Minkowski planes

AbstractGiven a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance difference of at most 2n ‐ 4, where n is the number of terminals. We show that this difference is best possible in the Euclidean plane, but not in Minkowski planes with parallelogram unit balls. We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete‐edge‐length condition. © 2010 Wiley Periodicals, Inc. NETWORKS, 2010

Positioning in a flat two-dimensional space-time: The delay master equation

The basic theory on relativistic positioning systems in a two-dimensional space-time has been presented in two previous papers [Phys. Rev. D {\bf 73}, 084017 (2006); {\bf 74}, 104003 (2006)], where the possibility of making relativistic gravimetry with these systems has been analyzed by considering specific examples. Here we study generic relativistic positioning systems in the Minkowski plane. We analyze the information that can be obtained from the data received by a user of the positioning system. We show that the accelerations of the emitters and of the user along their trajectories are determined by the sole knowledge of the emitter positioning data and of the acceleration of only one of the emitters. Moreover, as a consequence of the so called master delay equation, the knowledge of this acceleration is only required during an echo interval, i.e., the interval between the emission time of a signal by an emitter and its reception time after being reflected by the other emitter. We illustrate these results with the obtention of the dynamics of the emitters and of the user from specific sets of data received by the user.

The theorem of Schur in the Minkowski plane

We prove a Lorentzian analogue of the theorem of Schur for spacelike (or timelike) curves in the Minkowski plane.

Existence and uniqueness of classical solutions of the Cauchy problem on nonglobally hyperbolic manifolds

We consider the Cauchy problem for the wave equation on the Misner space, a nonglobally hyperbolic manifold with closed timelike lines. We prove that the existence and uniqueness of a classical solution are equivalent to self-consistency conditions much more rigorous than a finite collection of pointlike conditions occurring in this problem on the Minkowski plane with an attached handle.

Symmetric Minkowski planes ordered by separation

In Karzel et al. (J. Geom. 99: 116–127, 2009) we introduced for a symmetric Minkowski plane \({ {\mathfrak M} := (P,\Lambda,{\mathfrak G}_1,{\mathfrak G}_2) }\) an order concept by the notion of an orthogonal valuation for the circles of Λ and showed that there is a one to one correspondence between the valuations and the halforderings of the accompanying commutative field. Here we consider an order concept which is based on the notion of separation for quadruples of concyclic points and establish the connections between these two notions. Our main result (cf. Theorem 3.3) states that these concepts are equivalent.

Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be. The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the L 1 L^1 case has drawn much less attention. In this note we prove two quantitative isoperimetric inequalities in the L 1 L^1 Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch.

Circle configurations in strictly convex normed planes

AbstractWe present extensions of results of P. J. Kelly [Amer. Math. Monthly 57: 677–678, 1950] on the so-called re-entrant property of circles in strictly convex normed (or Minkowski) planes, and also further properties of circles, well known for the Euclidean plane, are generalized for all strictly convex Minkowski planes. More precisely, we present “Minkowskian analogues” of the philosophical symbol Yin-Yang, of the Arbelos, a special case of the famous Apollonius problem on circles touching each other, and one of the Sangaku-circles problems (coming from the Japanese Temple Geometry). The latter is remarkable since the consideration of this type of problems in the Euclidean plane requires the use of inversion or of the Pythagorean Theorem (i.e., of tools having no analogues in normed planes). Finally we observe that a strictly convex normed plane which, in addition, is smooth can be considered as a flat Möbius plane where, for example, Apollonius' problem is solved.

Restricted curvature in the Minkowski plane

A geometric proof for the following problem is given: Let E be the unit circle in a Minkowski plane. Let C be any continuously differentiable closed curve with length l(C) (measured in Minkowski metric). Assume |κe(C,.)| kκe(E,.) and κe(E,.) denote Euclidean curvatures. Then C can be contained in a similar copy of the unit disk translated and magnified by a factor of l(C)

Transitive groups of axial homologies of hyperbola structures and Minkowski planes

In this paper a typification of the automorphism groups of hyperbola structures based on the notion of axial homologies (i.e. automorphisms fixing two generators of the same kind) is given. For the class of hyperbola structures over half-ordered fields the types of the full automorphism groups are determined.

Large triangles contained in the unit disk of a Minkowski plane

We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are \({\frac{3}{2}}\) . We also prove that C permits to inscribe a triangle whose sides are of lengths at least \({\frac{3}{2}}\) in the positive orientation, or that they are of lengths at least \({\frac{3}{2}}\) in the negative orientation. The ratio \({\frac{3}{2}}\) in both the theorems is best possible.

Visibility in crowds of translates of a centrally symmetric convex body

Let B p , B q be disjoint translates of a centrally symmetric convex body B in R n . A translate B r of B lies between B p and B q if it overlaps none of B p and B q and there are points x ∈ B p , y ∈ B q such that the segment [ x , y ] meets B r . For a family F of translates of B lying between B p and B q and forming a packing, these two bodies are said to be visible from each other in the system { B p , B q } ∪ F if there exist points x , y like above such that [ x , y ] intersects no translate of F ; otherwise B p and B q are concealed from each other by F . The concealment number of a Minkowski space with unit ball B is the infimum of λ > 0 with ‖ p − q ‖ > λ implying that B p , B q can be concealed from each other by translates of B . Continuing the investigations of other authors, we prove several results about concealment numbers of Minkowski planes.

Reflections in strictly convex Minkowski planes

It is known that translations, symmetries with respect to points, and the identity map are the only isometries in general (normed or) Minkowski planes. Inspired by this “difference” to the Euclidean situation, we introduce so-called left-reflections in lines for the case of strictly convex Minkowski planes, and we develop a little theory on their products, yielding also results on glide reflections. As natural consequences we obtain several new characterizations of special normed planes, such as Radon planes or the Euclidean plane. All properties of left-reflections in strictly convex normed planes derived here hold in an analogous manner for correspondingly defined right-reflections in smooth Minkowski planes.

Covering Discs in Minkowski Planes

AbstractWe investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by k unit circles. In particular, we study the cases k = 3, k = 4, and k = 7. For k = 3 and k = 4, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, d-segments, and the monotonicity lemma.

Minimum Chords in Minkowski Planes

We extend the notion of minimum chords of circles from Euclidean geometry to the geometry of real two dimensional Banach spaces, also called Minkowski planes. In terms of unit discs, we first discuss the existence and uniqueness of minimum chords with respect to a given point of a Minkowskian disc. Then we study properties of the set of points of a Minkowskian disc whose minimum chords have lengths not less than a given positive number. Furthermore, the relations between the norms of such points and the lengths of corresponding minimum chords are investigated. New characterizations of the Euclidean plane are also derived.

On solutions to the wave equation on a non-globally hyperbolic manifold

We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of special form (the Minkowski plane with a handle) containing closed time-like curves. We prove that the classical solution of the Cauchy problem exists and is unique for initial data satisfying a specific set of additional requirements.

Total Angle around a Point in Minkowski Plane

A new angular measure in Minkowski plane was introduced recently. It is additive, invariant under invertible linear transformations, and has a clear interpretation as an "amount of rotation" from one direction to another one. The total angle τ around a point depends on the unit ball. It is known that \(4.443 \approx \sqrt{2}\pi \leq \tau \leq 8\). We show that \(4.985 \approx 4 \sqrt{2}\, {\rm ln}\,{\rm tan} \frac{3\pi}{8} \leq \tau \leq 8\, {\rm ln}\,{\rm tan} \frac{3\pi}{8}\, \approx 7.051\).