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Lipschitz constants and quadruple symmetrization by Möbius transformations

Oona Rainio, Matti Vuorinen

Abstract

Due to the invariance properties of cross-ratio, Möbius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a Möbius transformation, the distortion of the Euclidean geometry is rarely considered. Here, we identify several cases where the distortion caused by this symmetrization can be measured in terms of the Lipschitz constant of the Möbius transformation in the Euclidean or the chordal metric.

Lipschitz constants and quadruple symmetrization by Möbius transformations

Abstract

Due to the invariance properties of cross-ratio, Möbius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a Möbius transformation, the distortion of the Euclidean geometry is rarely considered. Here, we identify several cases where the distortion caused by this symmetrization can be measured in terms of the Lipschitz constant of the Möbius transformation in the Euclidean or the chordal metric.
Paper Structure (6 sections, 13 theorems, 58 equations, 4 figures)

This paper contains 6 sections, 13 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz constants in chordal metric
  4. Möbius symmetrization
  5. Three normalizations of point pairs
  6. Quadruples of points on the unit circle

Key Result

Theorem 1.1

Let $h$ be the Möbius transformation of $\overline{\mathbb{R}}^2$ that maps distinct points $a,b,c,d\in\mathbb{R}^2$ onto $-1,y,-y,1$, respectively. If $a,b,c,d$ are collinear and $|a-b|=|c-d|$, then $h^{-1}(0)=(a+d)/2$. Otherwise, where $y$ is as in Lemma lem_symth.

Figures (4)

  • Figure 1: The points $a,b,a^*,b^*$ when $a=-0.7i$ and $b=0.5$, the intersection point $c={\rm LIS}[a,b,a^*,b^*]$ of the lines $L(a,b)$ and $L(a^*,b^*)$, the unit circle, the circle $S^1(c,r)$ for $r=\sqrt{|c|^2-1}$, and the hyperbolic line $J^*[a,b]$. The end points of $J^*[a,b]$ are collinear with $c$, as denoted with dashed line. The Möbius transformation $T_{a,b}$ is a reflection over the line $L(0,b)$ composed with the inversion in the circle $S^1(c,r)$.
  • Figure 2: The Möbius transformation $T_k$ maps the nearest point $k$ on the hyperbolic line $J^*[a,b]$ through $a=-0.9i$ and $b=0.5-0.3i$ to the origin so that the images of $a$ and $b$ are collinear with the origin.
  • Figure 3: The Möbius transformation $f=T_{h_{ab},k}$ maps the hyperbolic midpoint $h_{ab}$ of $a$ and $b$ to the nearest point $k$ on the hyperbolic line $J^*[a,b]$ through $a=-0.9i$ and $b=0.5-0.3i$ so that the images of $a$ and $b$ are at the same distance from the origin.
  • Figure 4: The points $w_4$ and $w_5$ defining the Möbius transformation $T_{w_4,w_5}$ of Theorem \ref{['quadruples']} when $a=1$, $b=e^{0.3i}$, $c=e^{1.5i}$, and $d=e^{2.1i}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.9
  • proof
  • Theorem 2.19
  • Conjecture 3.1
  • ...and 11 more