Möbius transformation

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Not to be confused with the Möbius transform (Möbius inversion).

Möbius transformations are named in honor of August Ferdinand Möbius. They are also called homographic transformations, linear fractional transformations, fractional linear transformations or bilinear transformations.

A Möbius transformation of the plane is a rational function of the form

$w(z):={\frac {az+b}{cz+d}},$ of one complex variable $z$, where the coefficients $a,b,c,d$ are complex numbers satisfying $ad-bc\neq 0$.

This may be represented in matrix form by using homogeneous coordinates, where the two components of a vector represent numerator and denominator of a fraction:

$\left({\begin{matrix}{\begin{array}{c}az+b\\cz+d\end{array}}\end{matrix}}\right)=\left({\begin{matrix}{\begin{array}{cc}a&b\\c&d\end{array}}\end{matrix}}\right)\left({\begin{matrix}{\begin{array}{c}z\\1\end{array}}\end{matrix}}\right).\,$

Since $ad-bc$ is the determinant of the matrix, the matrix representing a Möbius transformation is invertible. Since scaling a numerator and denominator by the same complex number doesn't change the result, matrices which are nonzero scalar multiples of each other represent the same Möbius transformation. Thus the Möbius transformations are isomorphic to the projective linear group $PGL(2,\mathbb {C} )$.

The Möbius transformation is conformal, i.e. shape-preserving, thus maps circles to circles and lines to lines.

Inverse Möbius transformation

We find the inverse transform using the matrix representation:

$\left({\begin{matrix}{\begin{array}{cc}a&b\\c&d\end{array}}\end{matrix}}\right)^{-1}\left({\begin{matrix}{\begin{array}{c}w\\1\end{array}}\end{matrix}}\right)={\frac {1}{ad-bc}}\left({\begin{matrix}{\begin{array}{cc}d&-b\\-c&a\end{array}}\end{matrix}}\right)\left({\begin{matrix}{\begin{array}{c}w\\1\end{array}}\end{matrix}}\right)$,

and since ${\frac {1}{ad-bc}}$ is a scalar factor multiplying both numerator and denominator, the same transformation is represented by

$\left({\begin{matrix}{\begin{array}{cc}d&-b\\-c&a\end{array}}\end{matrix}}\right)\left({\begin{matrix}{\begin{array}{c}w\\1\end{array}}\end{matrix}}\right)=\left({\begin{matrix}{\begin{array}{c}dw-b\\-cw+a\end{array}}\end{matrix}}\right)$.

So $z(w)={\frac {dw-b}{-cw+a}}$.

Integer sequences

If iterating a given Möbius transformation yields an integer sequence, it is periodic with order 1, 2, 3, 4, or 6.

External links

• Möbius Transformations