Application of projective transformations preserving circles

A projective transformation (also called a homography) is a mapping of the projective plane that sends lines to lines. However, in general, projective transformations do not preserve circles. There is a special class of projective transformations, called Möbius transformations (or inversive/projective transformations preserving the circle), that do map circles to circles (or possibly lines, which can be thought of as circles through infinity).

Important properties

• A general projective transformation can send a circle to an ellipse, parabola, or hyperbola, but Möbius transformations send circles to circles (or lines).

• Möbius transformations can be written as $z \mapsto \frac{az + b}{cz + d}$ (for complex numbers $a, b, c, d$ with $ad - bc \neq 0$), and they preserve the set of all circles and lines in the complex plane.

• In geometry problems, using a suitable projective transformation that preserves circles can simplify configurations involving circles, tangency, and angles.