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# Möbius 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

which may be represented in matrix notation as

of one complex variable , where the coefficients are complex numbers satisfying

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

which may be represented in matrix notation as

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

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