Abstract In this paper we prove that there is a flock in every locally compact 2-dimensional Laguerre plane, i.e., a set of disjoint circles which covers the point space. We then determine the homeomorphism type of these flocks and use the relation between circle planes and generalised quadrangles to obtain analogous results about generalised quadrangles, Möbius planes and Minkowski planes.

We introduce a new family of field modes in flat spacetime—termed κ-—constructed from κ-dependent linear combinations of Minkowski plane waves. These modes define a one-parameter family of vacua |0κ⟩ that smoothly interpolate between different quantizations, reducing to the Minkowski vacuum in the limit κ→0. We show that |0κ⟩ is uniquely characterized as a continuous-mode squeezed vacuum, with frequency-dependent squeezing parameter r(ν) satisfying tanhr(ν)=e−πν/κ. We also derive two Bogoliubov transformations between κ-plane wave and κ-Rindler operators, which exhibit a universal form and smoothly interpolate between all known mode decompositions, including those of Minkowski, Rindler, and Unruh quantizations as limiting cases.

Hyperbola structures, Minkowski planes and Klein–Kroll type E

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

In 1947, Busemann observed that a Minkowski circle need not be a solution of the isoperimetric problem in a Minkowski plane. Li and Mo recently showed that the Euclidean circles centred at the origin in a unit ball with the Funk metric are solutions of the isoperimetric problem [9]. In this paper, we construct a class of Randers planes in which \textit{any} Euclidean circle, centered at the origin in ${\mathbb{R}}^2$, turns out to be a local minimum of the isoperimetric problem with respect to the various well-known volume forms in Finsler geometry. As a consequence, it turns out that the Euclidean circles centred at the origin are solutions of the isoperimetric problem in a Randers type Minkowski plane.

Semi-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.

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.

We consider the four-dimensional Lie group SE(1, 1) - a connected group of motions of the Minkowski plane that preserve the orientation of the plane. A unique, up to isometry, left-invariant Lorentzian metric is found, together with which the given Lie group is a self-similar manifold. Formulas are obtained that describe the action of an essentialone-parameter similarity group with respect to natural coordinates associated with the matrix representation of the Lie group. This similarity group is generated by the oneparameter self- similarity group of the corresponding Lie algebra, equipped with a Lorentz scalar product.

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.

In this paper, we extend the method developed in [17, 18] to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in detail with all local phenomena that occur generically in 2-parameters families of curves. In each case, we obtain the geometry of the deformed curve, that is, information about inflections, vertices and lightlike points. We also obtain the behavior of the evolute/caustic of a curve at especial points and the bifurcations that can occur when the curve is deformed.

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.

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).

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.

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 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.

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

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

<abstract><p>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.</p></abstract>