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Linear Fractional Transformation


A transformation of the form

 w=f(z)=(az+b)/(cz+d),
(1)

where a, b, c, d in C and

 ad-bc!=0,
(2)

is a conformal mapping called a linear fractional transformation. The transformation can be extended to the entire extended complex plane C^*=C union {infty} by defining

f(-d/c)=infty
(3)
f(infty)=a/c
(4)

(Apostol 1997, p. 26). The linear fractional transformation is linear in both w and z, and analytic everywhere except for a simple pole at z=-d/c.

Kleinian groups are the most general case of discrete groups of linear fractional transformations in the complex plane z->(az+b)/(cz+d).

Every linear fractional transformation except f(z)=z has one or two fixed points. The linear fractional transformation sends circles and lines to circles or lines. Linear fractional transformations preserve symmetry. The cross ratio is invariant under a linear fractional transformation. A linear fractional transformation is a composition of translations, rotations, magnifications, and inversions.

To determine a particular linear fractional transformation, specify the map of three points which preserve orientation. A particular linear fractional transformation is then uniquely determined. To determine a general linear fractional transformation, pick two symmetric points alpha and alpha_S. Define beta=f(alpha), restricting beta as required. Compute beta_S. f(alpha_S) then equals beta_S since the linear fractional transformation preserves symmetry (the symmetry principle). Plug in alpha and alpha_S into the general linear fractional transformation and set equal to beta and beta_S. Without loss of generality, let c=1 and solve for a and b in terms of beta. Plug back into the general expression to obtain a linear fractional transformation.


See also

Cayley Transform, Kleinian Group, Möbius Transformation, Modular Group Gamma, Schwarz's Lemma, Symmetry Principle, Unimodular Transformation

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References

Anderson, J. W. "The Group of Möbius Transformations." §2.1 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 19-25, 1999.Apostol, T. M. "Möbius Transformations." Ch. 2.1 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 26-28, 1997.Krantz, S. G. "Linear Fractional Transformations." §6.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 81-86, 1999.Mathews, J. "The Moebius Transformation." http://www.ecs.fullerton.edu/~mathews/fofz/mobius/.Needham, T. "Möbius Transformations and Inversion." Ch. 3 in Visual Complex Analysis. New York: Clarendon Press, pp. 122-188, 2000.

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Linear Fractional Transformation

Cite this as:

Weisstein, Eric W. "Linear Fractional Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LinearFractionalTransformation.html

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