Mapping Properties of a Scale Invariant Cassinian Metric and a Gromov Hyperbolic Metric

Abstract

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.2010 Mathematics subject classification: primary 51M10; secondary 26A15, 30C20, 30C65, 30F45.

Approximation of Generalized Ovals and Lemniscates towards Geometric Modeling

“…The building of symmetric closed curves with two or more foci, i.e., figures like ellipses, Perseus curves, Cayley ovals, and lemniscates, is of particular practical interest [14][15][16][17][18]. Some of these curves are well studied, such as special cases of Perseus curves, Cassini ovals, Bernoulli lemniscate, and Booth lemniscate, and are applied to radiolocation, group theory, cluster analysis, engineering and construction design, and quantum physics [19][20][21][22][23][24][25]. Less investigated are, for example, Cayley ovals, which find application in connection with studies of elementary particle trajectories [26][27][28].…”

A Gromov Hyperbolic metric and Möbius transformations

“…Indeed, Gehring, Palka, and Osgood have proved that the quasihyperbolic metric and the distance ratio metric are not changed by more than a constant 2 under Möbius transformations, see [GP,Corollary 2.5] and [GO,proof of Theorem 4]. Several authors have also studied this topic for other hyperbolic type metrics in [CHKV,HVZ,I3,KLVW,MS1,MS2,SVW,WV,XW].…”

“…Recently, Ibragimov introduced a new metric u Z to hyperbolize the locally compact noncomplete metric space (Z, d) without changing its quasiconformal geometry which is defined as [I2] Several authors have studied comparison inequalities between the Gromov hyperbolic metric and the hyperbolic metric as well as some hyperbolic type metrics [I2,MS1,Z]. Mohapatra and Sahoo also investigated quasi-invariance properties of the Gromov hyperbolic metric under quasiconformal mappings [MS2].…”

Remarks on the scale invariant Cassinian metric

“…If we can describe the balls of a metric space "explicitly", then we already know a lot about the geometry of the metric -this requires that we can estimate the metric in terms of well-known metrics. For a survey and comparison inequalities between some of these metrics, see [4,7,10,15,16,17,19,20].…”

“…It is readily seen that the metric τD is invariant under similarity transformations. Several authors [12,15,16] have studied some basic properties of the scale invariant Cassinian metric and its distortion under Möbius transformations of the unit ball, and also quasiinvariance properties under quasiconformal mappings.…”