This site is supported by donations to The OEIS Foundation.

# Möbius transformation

(Redirected from Homographic transformation)

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

${\displaystyle w(z):={\frac {az+b}{cz+d}},}$

which may be represented in matrix notation as

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

of one complex variable ${\displaystyle z}$, where the coefficients ${\displaystyle a,b,c,d}$ are complex numbers satisfying

${\displaystyle \left|{\begin{matrix}{\begin{array}{cc}a&b\\c&d\end{array}}\end{matrix}}\right|=ad-bc\neq 0,}$

so that the matrix is invertible.

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

## Inverse Möbius transformation

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

which may be represented in matrix notation as

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

if and only if (Verify: it seems that det ≠ 0 is not enough, we need det = 1.)[1]

${\displaystyle \left|{\begin{matrix}{\begin{array}{cc}a&b\\c&d\end{array}}\end{matrix}}\right|=ad-bc=1.}$

## Integer sequences

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

## Notes

1. Needs verification (it seems that det ≠ 0 is not enough, we need det = 1).
2. Donald M. Adelman, Note on the arithmetic of bilinear transformations, Proc. Amer. Math. Soc. 1 (1950), pp. 443-448.