Concepts of curvatures in normed planes

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.     

On positively differentiable curves with values in normed linear spaces

We prove a theorem that characterizes continuous normed linear space-valued curves allowing differentiable parameterizations with non-zero derivatives as those curves, all the points of which are regular (in Choquet's sense). We also state an equivalent geometric condition not involving any homeomorphisms. This extends a theorem due to Choquet, who proved a similar result for curves with values in Euclidean spaces.     

A new construction of Radon curves and related topics

We present a new construction of Radon curves which only uses convexity methods. In other words, it does not rely on an auxiliary Euclidean background metric (as in the classical works of J. Radon, W. Blaschke, G. Birkhoff, and M. M. Day), and also it does not use typical methods from plane Minkowski Geometry (as proposed by H. Martini and K. J. Swanepoel). We also discuss some properties of normed planes whose unit circle is a Radon curve and give characterizations of Radon curves only in terms of Convex Geometry.     

Pedal curves of frontals in the Euclidean plane

In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.     

An orthogonality in normed linear spaces based on angular distance inequality

In this paper, we present a new orthogonality in a normed linear space which is based on an angular distance inequality. Some properties of this orthogonality are discussed. We also find a new approach to the Singer orthogonality in terms of an angular distance inequality. Some related geometric properties of normed linear spaces are discussed. Finally a characterization of inner product spaces is obtained.     

Pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz–Minkowski space

In this paper we introduce the notion of pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz?CMinkowski space which is analogous to the notion of evolutes of curves on the hyperbolic plane. We investigate the singularities and geometric properties of pseudo-spherical evolutes of curves on a spacelike surface.     

Normal form for space curves in a double plane

This note is an attempt to relate explicitly the geometric and algebraic properties of a space curve that is contained in some double plane. We show in particular that the minimal generators of the homogeneous ideal of such a curve can be written in a very specific form. As applications we characterize the possible Hartshorne–Rao modules of curves in a double plane and the minimal curves in their even Liaison classes.     

A new type of orthogonality for normed planes

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ⩾ 3.     

Geometry of root-related parameters of PH curves

We study the (plane polynomial) Pythagorean hodograph curves from the viewpoint of their roots. The loci of root-related parameters of PH curves show us very interesting geometric properties. They include regular 2n + 1-gon and isosceles triangles with the ratio of sides n : 1 : n.     

On Miquel’s theorem and inversions in normed planes

Asplund and Grünbaum proved that Miquel’s six-circles theorem holds in strictly convex, smooth normed planes if the considered circles have equal radii. We extend this result in two directions. First we prove that Miquel’s theorem for circles of equal radii (more precisely, a generalized version of it) is true in every normed plane, without the assumptions of strict convexity and smoothness, and give also some properties of the circle configuration related to this theorem. Second we clarify the situation if the circles of the corresponding configuration do not necessarily have equal radii.     

The Plane Sextic Curve Fixed by A6

Plane curves with non-trivial collineation groups are rare: those of low order are thus interesting, and often exhibit special geometric features. The largest primitive plane group is A₆. It is known by standard algebraic means that this fixes a sextic curve Ω. The present paper constructs Ω geometrically, and speedily obtains its most significant geometric property: its 72 inflexions lie by pairs on 36 biflexional tangents. There is no standard technique for determining the multiplicities of bitangents of plane curves. For Ω we show that each biflexional tangent counts 4-fold as a bitangent, and identify the other 180 ordinary bitangents. A brief comparison of the geometric properties of Ω with those of Klein’s quartic curve is given.     

椭圆—卡西尼卵形线

给出平面上到两个定点距离的调和平均值等于定值的点所满足的方程及其图形 ,称为椭圆—卡西尼卵形线 ,得到它的一些性质 ,以及其极值点所满足的方程与图形     

Curves on the shape sphere

Using a special conformai map between the two-dimensional sphere and the extended plane, we describe some classes of curves on the sphere. We also discuss a differential geometric invariant determining a plane curve up to a direct similarity and study self-similar plane curves.     

Variational grid generation

Recently, variational methods have been used to numerically generate grids on geometometric objects such as plane regions, volumes, and surfaces. This article presents a new method of determining variational problems that can be used to control such properties of the grid as the spacing of the points, area or volume of the cells, and the angles between the grid lines. The methods are applied to curves, surfaces, and volumes in three-dimensional space; then segments, plane curves, and plane regions appear as special cases of the general discussion. The methods used here are simpler and clearer and provide more direct control over the grid than methods that appear elsewhere. The methods are applicable to any simply connected region or any region that can be made simply connected by inserting artificial boundaries. The methods also generalize easily to solution-adaptive methods. An important ingredient in our method is the notion of a reference grid. A reference grid is defined on a region that is simpler, but analogous to, the geometric object on which a grid is desired. Variational methods are then used to transfer the reference grid to the geometric object. This gives simple and precise control of the local properties of the grid.     

Projections,Birkhoff Orthogonality and Angles in Normed Spaces

Let $X$ be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle $A_q(x, y)$ between two vectors $x$ and $y$ in $X$, such that $x$ is Birkhoff orthogonal to $y$ if and only if $A_q(x, y) =\frac{π}{2}$. Some other properties of this angle are also discussed.     

赋范BCK-代数(英文)

引入了BCK-代数的范数与距离的概念,给出了赋范BCK-代数的一些基本性质,证明了赋范BCK-代数的同构(同态)像和原像仍是赋范BCK-代数,研究了BCK-代数与BCK-代数笛卡儿之间的赋范性质关系.并且引入了赋范BCK-代数的点列极限概念,研究了极限的相关性质.讨论了有界赋范BCK-代数的与模糊BCK-代数的关系.     

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.     

Generic equi-centro-affine differential geometry of plane curves

We study the equi-centro-affine invariants of plane curves from the view point of the singularity theory of smooth functions. We define the notion of the equi-centro-affine pre-evolute and pre-curve and establish the relationship between singularities of these objects and geometric invariants of plane curves.     

Continuities of Metric Projection and GeometricConsequences

We discuss the geometric characterization of a subsetKof a normed linear space via continuity conditions on the metricprojection ontoK. The geometric properties considered includeconvexity, tubularity, and polyhedral structure. The continuityconditions utilized include semicontinuity, generalized stronguniqueness and the non-triviality of the derived mapping. Infinite-dimensional space with the uniform norm we show thatconvexity is equivalent to rotation-invariant almost convexityand we characterize those sets every rotation of which has continuousmetric projection. We show that polyhedral structure underliesgeneralized strong uniqueness of the metric projection.