Related papers: Classifying Isometries
We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of…
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.…
We prove that any isometry between two dimensional Hilbert geometries is a projective transformation unless the domains are interiors of triangles.
We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of…
Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous…
This paper is devoted to the study of isometrically homogeneous spaces from the view point of metric geometry. Mainly we focus on those spaces that are homeomorphic to lines. One can reduce the study to those distances on $\R$ that are…
Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar…
A stratified space is a topological space equipped with a \emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e.,…
The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.
In this paper we consider the isoptic curves on the 2-dimensional geometries of constant curvature $\bE^2,~\bH^2,~\cE^2$. The topic is widely investigated in the Euclidean plane $\bE^2$ see for example \cite{CMM91} and \cite{Wi} and the…
This note is devoted to proving the following result: given a compact metrizable group G, there is a compact metric space K such that G is isomorphic (as a topological group) to the isometry group of K.
The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic…
We introduce a generalization of the b-metric we call a (b,c)-metric. We show that if $X$ is a $(b,c)$-metric space and $\psi: X \longrightarrow Y$ is a quasi-isometry then $Y$ is $(b,c)$-metrizable. We also define a particular kind of…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the…
We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal…
A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.
The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which…
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an "isogeny estimate", providing an explicit…
Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the…