Conformal Mapping & Bilinear Transformations Cheat Sheet

Conformal mapping transforms regions in the complex plane while preserving angles. Bilinear transformations are a specific type of mapping that preserve circles and lines, crucial for simplifying complex problems.

Core Principles

Conformal mapping preserves angles (magnitude and sense) between intersecting curves.
A function f(z) is conformal where it is analytic and f'(z) ≠ 0.
The angle of rotation is arg(f'(z)) and the scale factor is |f'(z)|.
Bilinear transformations (w = (az+b)/(cz+d)) map circles and lines to circles and lines.
Bilinear transformations preserve the cross-ratio of four points.
Fixed points of a transformation are where f(z) = z.

Action Steps

Identify the type of transformation (translation, rotation, magnification, inversion, bilinear).
Determine how basic shapes (lines, circles) transform under the given mapping.
Calculate the image of specific points or regions by substituting into the transformation equation.
For conformal mappings, find f'(z) to determine angle of rotation and scale factor.
For bilinear transformations, use the cross-ratio property or map three points to find the transformation.

Formulas

Angle of rotation: $\psi = \arg(f'(z))$
Scale factor: $|f'(z)|$
Bilinear transformation: $w = \frac{az+b}{cz+d}$, where $ad-bc \neq 0$
Inverse bilinear transformation: $z = \frac{-dw+b}{cw-a}$
Cross-ratio: $\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} = \frac{(w_1-w_3)(w_2-w_4)}{(w_1-w_4)(w_2-w_3)}$
Fixed points: $f(z) = z$

Key Terms

Conformal Mapping: A transformation that preserves angles locally.
Bilinear Transformation: A transformation of the form w = (az+b)/(cz+d) that maps circles/lines to circles/lines.
Fixed Point: A point z such that f(z) = z.
Critical Point: A point where f'(z) = 0.
Cross-Ratio: An invariant quantity under bilinear transformations.
Translation: w = z + c
Rotation: w = ze^{i\phi}
Magnification: w = bz (b is real)
Inversion: w = 1/z

Real World Examples

Fluid dynamics: Conformal mapping simplifies problems by transforming complex flow regions into simpler ones.
Electrostatics: Mapping can be used to find electric potential in complex geometries.
Cartography: Map projections are related to conformal transformations.