ConformalParameterization - crowlogic/arb4j GitHub Wiki

A conformal parameterization of the complex plane is a way of mapping the complex plane onto a domain in the complex plane such that angles between curves are preserved. In other words, a conformal parameterization is a mapping that preserves the local geometry of the complex plane.

One example of a conformal parameterization of the complex plane is the exponential map, which maps the complex plane onto the punctured complex plane (i.e., the complex plane with the origin removed) by sending each point z to the point exp(z). This map is conformal because it preserves angles between curves. For example, the lines x = c (where c is a constant) in the complex plane are mapped to circles centered at the origin in the punctured complex plane, and the angles between these lines are preserved under the exponential map.

Other examples of conformal parameterizations of the complex plane include the Mobius transformations, which are mappings of the form f(z) = (az + b)/(cz + d), where a, b, c, and d are complex constants and ad - bc ≠ 0. These mappings can be used to map the complex plane onto various domains in the complex plane, such as circles, half-planes, and annuli, while preserving angles between curves.