Projective transformations of the line

A projective transformation of the line is a special way of mapping points on a line to other points on the same (or another) line, so that the cross ratio of any four points is preserved. These transformations are also called homographies or Möbius transformations (in the real or complex case).

Important properties

• Projective transformations send points on the projective line to other points on the projective line.

• They preserve the cross ratio of four points.

• Any projective transformation can be written as $x \mapsto \dfrac{ax + b}{cx + d}$, where $a, b, c, d$ are real numbers and $ad - bc \neq 0$.

• They include translations, scalings, and more general mappings.