Conformal Moduli of Symmetric Circular Quadrilaterals With Cusps

TL;DR

This paper develops analytic and numerical methods to compute the conformal moduli of symmetric circular quadrilaterals with cusps, validating results through multiple approaches and providing explicit examples.

Contribution

It introduces an analytic reduction to ODEs and employs two numerical methods, Schwarz equation and hpFEM, for accurate computation of conformal moduli of symmetric circular quadrilaterals.

Findings

Validated numerical methods with high accuracy

Derived explicit conformal moduli for specific quadrilaterals

Established agreement between different computational approaches

Abstract

We investigate moduli of planar circular quadrilaterals symmetric with respect to both the coordinate axes. First we develop an analytic approach which reduces this problem to ODEs and devise a numeric method to find out the accessory parameters. This method uses the Schwarz equation to determine conformal mapping of the unit disk onto a given circular quadrilateral. We also give an example of a circular quadrilateral for which the value of the conformal modulus can be found in the analytic form; this example is used to validate the numeric calculations. We also use another method, so called hpFEM, for the numeric calculation of the moduli. These two different approaches provide results agreeing with high accuracy.