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Gears, Pregears and Related Domains

Philip R. Brown, R. Michael Porter

Abstract

We study conformal mappings from the unit disk to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or "gearlike") domains fit into a more general category of domains we call "pregears" (images of gears under Mobius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.

Gears, Pregears and Related Domains

Abstract

We study conformal mappings from the unit disk to one-toothed gear-shaped planar domains from the point of view of the Schwarzian derivative. Gear-shaped (or "gearlike") domains fit into a more general category of domains we call "pregears" (images of gears under Mobius transformations), which aid in the study of the conformal mappings for gears and which we also describe in detail. Such domains being bounded by arcs of circles, the Schwarzian derivative of the Riemann mapping is known to be a rational function of a specific form. One accessory parameter of these mappings is naturally related to the conformal modulus of the gear (or pregear) and we prove several qualitative results relating it to the principal remaining accessory parameter. The corresponding region of univalence (parameters for which the rational function is the Schwarzian derivative of a conformal mapping) is determined precisely.

Paper Structure

This paper contains 10 sections, 12 theorems, 35 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Schwarzian derivative and accessory parameters
  3. Gear domains
  4. The Schwarzian derivative of a gear mapping
  5. Conformal modules of gear domains
  6. The region of gearlikeness
  7. Pregear domains
  8. Degenerate gears and pregears
  9. The boundary of the region of gearlikeness
  10. Discussion and conclusions

Key Result

Proposition 2.1

A necessary and sufficient condition for $f(z)$ to be a univalent mapping of the unit disk onto a one-tooth gear domain, satisfying the normalizations $f(0)=0$ and where the prevertices $e^{\pm it_1}$ map to the vertices with interior angles $\pi/2$ and the prevertices $e^{\pm it_2}$ map to the vertices with interior angles $3\pi/2$, is that Furthermore, the gear ratio $\beta$ and the gear angle

Figures (6)

  • Figure 1: Gear parameters.
  • Figure 2: Examples of pregears. The circles $C^\pm$ containing the tooth edges are solid gray; the circles containing the A- and B-arcs in dotted gray.
  • Figure 3: The two types of degenerate pregears
  • Figure 4: Extreme cases of degenerate pregears, for $\eta\to0$ (above) and $\eta\to\infty$ (below).
  • Figure 5: Region of gearlikeness (gray, bounded by solid curves, which correspond to degenerate gears).
  • ...and 1 more figures

Theorems & Definitions (17)

  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Conjecture 3.5
  • Theorem 3.6
  • ...and 7 more