public class AlgorithmNearlyCircleToCircle
extends AlgorithmBase

This program performs the conformal mapping of a nearly circular region to a circle. The conformal mapping is done in 3 steps: 1.) Translate the circle from the center of the square image to the origin and convert to a unit disc by dividing the distance from the center by the radius. 2.) Map from the unit circle to a unit nearly circular region at the origin. 3.) Map from the unit nearly circlular region at the origin to the original nearly circular region by scaling and translating.

References: 1.) Advanced Calculus For Applications Second Edition by F. B. Hildebrand, Section 10.4 Analytic Functions of a Complex Variable pages 550-554 and Section 11.4 Conformal Mapping pages 628-632, Prentice-Hall, Inc., 1976. 2.) Conformal Mapping by Zeev Nehari, Dover Publications, Inc., 1952, Chapter V, Conformal Mapping of Simply Connected Domains, Section 11, Conformal Mapping of Nearly Circular Domains, pp. 263 - 265. 3.) "On the Conformal Mapping of Nearly-Circular Domains" by Zeev Nehari and Vikramaditya Singh, Proceedings of the American Mathematical Society, 1956, pp. 370-378. According to the Nehari book: The function: F(z) = z + ((eps*z)/(2*PI))*Integral from 0 to 2*PI of ((exp(i*theta) + z)/(exp(i*theta) - z))*p(theta)d(theta) maps abs(z) 0. Multiply both the numerator and the denominator of the integral by (exp(i*theta) - z)* = (exp(-i*theta) - z*) to create a real denominator denom = 1 - 2xcos(theta) - 2ysin(theta) + x**2 + y**2 The real part of the integral equals: Integral from 0 to 2*PI of ((1 - x**2 - y**2)/denom)p(theta)d(theta) The imaginary part of the integral equals: Integral from 0 to 2*PI of ((-2xsin(theta) + 2ycos(theta))/denom)p(theta)d(theta) 0 to 2*PI direction would be counterclockwise. Suppose curve is clockwise, then the theta increments are negative. So both a counterclockwise integral and the negative of the clockwise integral are realized by taking the absolute value of the theta angle increment.