Conformal Mappings

An Application on Cryptography

1. Introduction

We introduce conformal mappings by a simple example. Enunciate an important theorem. Show the most common conformal transformations.

A complex number (where ) is plotted in the Argand plane with real component x along the horizontal axis and imaginary component y along the vertical axis.

Consider a complex function . Because a full Cartesian plot of requires four real dimensions, we visualize f as a geometric mapping from the domain z-plane to the target w-plane.


Let . Expanding in rectangular coordinates:

         (1.1)

Equating real and imaginary components gives:

          (1.2)

Computing the complex derivative:

              (1.3)



Since f(z) is analytic and , f is a conformal mapping that preserves both the magnitude and orientation of angles between intersecting curves. In this specific case, the action is a rigid translation of the coordinate grid by the vector (1, 1).

The following script can be run in Wolfram’s Mathematica to plot the transformation f domain and codomain.

We observe that the target plane is a rigid translation of the domain plane by (1, 1). Because the derivative of the mapping is everywhere (and its real Jacobian is the identity matrix), any smooth curve in the z-plane maps to in the w-plane such that their tangent vectors are identical:
                                                       (1.4)