Möbius Transformations

Complex Analysis · Axiom Academy

The most elegant maps in complex analysis — and they bend straight lines into perfect circles. A whole family of maps, built from one rule A Möbius transformation (also called a fractional linear transformation) is any map of the extended complex plane of the form below. Just four complex numbers, one ratio — yet this family turns out to be the natural language for geometry on the sphere, hyperbolic space, and even relativity. Before any formulas, watch the one property that makes these maps special. A straight line is fed through the simplest Möbius map, the inversion w = 1/z . Watch the line bow until it closes up into a perfect circle through the origin . Lines and circles trade places — but the family of "circles-and-lines" is preserved. Möbius transformations are the only maps that always send circles-and-lines to circles-and-lines. Slide the line — read off its image circle Here is the vertical line in blue on the left, and its image under w = 1/z in orange on the right. Drag the slider (or the line itself) to move a . The image is always a circle through the origin with center and radius — watch the readout confirm it as you move. Push the line toward the imaginary axis ( ) and the circle balloons toward a line. A line through no special point becomes a circle; the only circle-and-line that stays a line is one through the origin. Every Möbius map is just four simple moves

This is the written version of the interactive lesson above. See the full Complex Analysis course.