Minkowski Plane Research Articles

The isoperimetric problem for convex hulls and the large deviations rate functionals of random walks

We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite, such a scaled limit trajectory h solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of h is replaced by the large deviations rate functional ∫01I(h′(t))dt and I is the rate function of the increments. Assuming that the distribution of increments is not supported on a half-plane, we show that the optimal trajectories are convex and satisfy the Euler–Lagrange equation, which we solve explicitly for every I. The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).

Semi-classical 2-dimensional Minkowski planes

AbstractSemi-classical geometries have been investigated in the context of 2-dimensional affine planes, projective planes, Möbius planes and Laguerre planes. Here we deal with the case of 2-dimensional Minkowski planes. Semi-classical 2-dimensional Minkowski planes are obtained by pasting together two halves of the classical real Minkowski plane along two circles or parallel classes. By solving some functional equations for the functions that describe the pasting we determine all semi-classical 2-dimensional Minkowski planes. In contrast to the rich variety of other semi-classical planes there are only very few models of such Minkowski planes.

Relationship between Generalized Orthogonality and Gâteaux Derivative

This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of x in the y direction is zero when the norm is Gâteaux differentiable in the y direction at x and x and y satisfy certain generalized orthogonality conditions. A case where x and y are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the ⊥Δ coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper.

Evolutoids and pedaloids of frontals on timelike surfaces

Abstract In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.

Geometric deformations of curves in the Minkowski plane

Geometric deformations of curves in the Minkowski plane

Note on the Chromatic Number of Minkowski Planes: The Regular Polygon Case

The famous Hadwiger–Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice–sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.

The Dunkl-Williams constant related to the singer orthogonality and red isosceles orthogonality in Banach spaces

In this paper, we shall consider two new constants DWS(X) and DWI(X), which are the Dunkl-Williams constant related to the Singer orthogonality and theisosceles orthogonality, respectively. We discuss the relationships between DWS(X) and some geometric properties of Banach spaces, including uniform non-squareness, uniform convexity. Furthermore, an equivalent form of DWS(X) in the symmetric Minkowski planes is given and used to compute the value of DWS((R2, ???p)), 1 < p < ?, and we also give a characterization of the Radon plane with affine regular hexagonal unit sphere in terms of DWS(X). Finally, we establish some estimates for DWI(X) and show that DWI(X) does not necessarily coincide with DWS(X).

A family of flat Minkowski planes over convex functions

Using suitable convex functions, we construct a new family of flat Minkowski planes whose automorphism groups are at least 3-dimensional. These planes admit groups of automorphisms isomorphic to the direct product of $${\mathbb {R}}$$ and the connected component of the affine group on $${\mathbb {R}}$$ . We also determine isomorphism classes, automorphisms and possible Klein–Kroll types for our examples.

Quadratic Hyperboloids in Minkowski Geometries

A Minkowski plane is Euclidean if and only if at least one hyperbola is a quadric. We discuss the higher dimensional consequences too.

Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane

In this paper, we take advantage of envelope theory and singularity theory to study the evolutoids and pedaloids in Minkowski plane. We illustrate an internal correlation from algebraic and geometric viewpoints, and give the geometric explanation of evolutoids and pedaloids. Then, we generalize the notions of evolutoids and pedaloids to the category of frontals in Minkowski plane. Furthermore, we apply the technique of singularity theory, using the discriminants and versal unfolding tools, to consider evolving evolutoids and give the classifications of singularities of the evolutoids, and explain when cusps and inflexions occur and how evolutoids evolving. Besides, we found that there is a close relationship between the evolutoids and pedaloids and good correspondence between their singularities.

Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz–Minkowski plane R12

In this paper, we investigate the evolution of spacelike curves in the Lorentz–Minkowski plane R12 along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike grim reaper curve as time tends to infinity.

Anti-pedals and Primitives of Curves in Minkowski Plane

The orthogonal projection of a fixed point on the tangent lines of a given curve yields a pedal curve of that curve. The aim of this study is to examine some special curves, such as pedal curves, which have singular points even for regular curves, in the Minkowski plane. For this, we investigate an anti-pedal and a primitive of curve, which is closely related to the pedal curve. The primitive of a curve is a curve that is provided by the inverse construction to make pedal. Using the envelope of a family of functions, we obtain the notion of primitive for the curves in the Minkowski plane. Then, we show that an anti-pedal of the original curve is equal to the inversion image of the pedal curve. Moreover, we analyze the relationships between primitive and anti-pedal of the curve using the inversion. We also present examples that provide our results.

Erdős-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields

In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles.As a corollary, we prove that the largest families of polynomials over Fq of degree at most k, with 2≤k<q, which pairwise take the same value on at least one point, consist of all polynomials f of degree at most k such that f(x)=y for some fixed x and y in Fq.We also discuss this problem for ovoidal Minkowski planes, and we investigate the largest families of circles pairwise intersecting in two points in circle geometries.

Topology of Integrable Billiard in an Ellipse in the Minkowski Plane with the Hooke Potential

Topology of Integrable Billiard in an Ellipse in the Minkowski Plane with the Hooke Potential

Primitivoids of curves in Minkowski plane

&lt;abstract&gt;&lt;p&gt;In this work, we investigate the differential geometric characteristics of pedal and primitive curves in a Minkowski plane. A primitive is specified by the opposite structure for creating the pedal, and primitivoids are known as comparatives of the primitive of a plane curve. We inspect the relevance between primitivoids and pedals of plane curves that relate with symmetry properties. Furthermore, under the viewpoint of symmetry, we expand these notions to the frontal curves in the Minkowski plane. Then, we present the relationships and properties of the frontal curves in this category. Numerical examples are presented here in support of our main results.&lt;/p&gt;&lt;/abstract&gt;

Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology

We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of these billiard systems in terms of Fomenko invariants. Then we provide periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases of billiards on the Minkowski and Euclidean planes. Dedicated to R. Baxter on the occasion of his 80th anniversary. Bibliography: 51 titles.

Lorentzian Approximations and Gauss–Bonnet Theorem for E(1,1) with the Second Lorentzian Metric

In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski plane . By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in E(1,1) with the second Lorentzian metric away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in E(1,1) with the second left‐invariant Lorentzian metric g2.

Liouville Foliation of Topological Billiards in the Minkowski Plane

In the paper, we give the Liouville classification of five interesting cases of topological billiards glued from two flat billiards bounded by arcs of confocal quadrics in the Minkowski plane. For each billiard, we calculate the marked Fomenko–Zieschang molecule, in other words the invariant of an integrable Hamiltonian system that completely determines the type of its Liouville foliation.

Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane R12. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.

Conformality: classical and generalized

We discuss several topics: the concept of conformal mapping of Riemannian and pseudo-Riemannian manifolds, conformal rigidity of higher-dimensional domains, and conformal flexibility of two-dimensional domains of Euclidian and Minkowski planes. We present an extension of the concept of conformal mapping proposed by M. Gromov and recall an open problem related to it.