Related papers: Proper quadrics in the Euclidean $n$-space
We study the concept of cone metric space in the context of ordered vector spaces by setting up a general and natural framework for it.
We study the duality of moduli of k- and (n-k)-dimensional slices of euclidean n-cubes, and establish the optimal upper bound 1.
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow.…
This article presents a new way to classify geodesics on a cone in the Euclidean 3-space. This proof is obtained considering our main result, which establishes the necessary and sufficient conditions that a curve in space must satisfy: to…
This paper is a continuation of the papers [2,3,4,5,6]. In this paper the osculating spaces of arbitrary order of a manifold embedded in Euclidean space are considered. A better estimation of their dimensions as well as the description of…
We introduce the {\em pair-cut cone $PCUT_n$} of metrics on sets with $n\ge 3$ elements, that correspond to linear combinations with non-negative coefficients of the cut-metrics resulting from cuts that are pairs. Given a metric, we fully…
We consider the quasihyperbolic metric, and its generalizations in both the $n$-dimensional Euclidean space $R^n$, and in Banach spaces. Historical background, applications, and our recent work on convexity properties of these metrics are…
Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding…
We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the…
In this work, notion of a slant helix is extended to space E$^n$. Necessary and sufficient conditions to be a slant helix in the Euclidean $n-$space are presented. Moreover, we express some integral characterizations of such curves in terms…
We determine explicit formulas for geodesics (in the Euclidean metric) in the configuration space of ordered pairs (x,x') of points in R^n which satisfy d(x,x')>=epsilon. We interpret this as two or three (depending on the parity of n)…
The article deals with plastic and non-plastic sub-spaces $A$ of the real line ${\mathbb{R}}$ with the usual Euclidean metric $d$. It investigates non-expansive bijections, proves properties of such maps and demonstrates their relevance by…
Sectors at centre of affine quadrics with point symmetry are investigated over arbitrary fields of characteristic different from two. As an application we demonstrate nice formulas for the area and the volume of such planar and spatial…
In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
In this paper we overview the theory of conics and roulettes in four non-Euclidean planes. We collect the literature about these classical concepts, from the eighteenth century to the present, including papers available only on arXiv. The…
We give a metric characterization of snowflakes of Euclidean spaces. Namely, a metric space is isometric to $\mathbb R^n$ equipped with a distance $(d_{\rm E})^\epsilon$, for some $n\in \mathbb N_0$ and $\epsilon\in (0,1]$, where $d_{\rm…
Embedding of the brane metric into Euclidean (2+4)-space is found. Brane geometry can be visualized as the surface of the hyper-sphere in six dimensions which 'radius' is governed by the cosmological constant. Minkowski space in this…