Linear Fractional Transformation

A linear fractional transformation is a transformation of the form

(1)

where , , , and

(2)

It defines a conformal mapping of the extended complex plane . For , the extension is given by

			(3)
			(4)

(Apostol 1997, p. 26). For , the transformation is and fixes infinity. For , it is an analytic function away from its simple pole at . The equation is linear in either variable when the other is held fixed.

Kleinian groups are discrete groups of linear fractional transformations of the extended complex plane.

Every linear fractional transformation except has one or two fixed points in the extended complex plane. A linear fractional transformation sends circles and lines to circles or lines, with infinity included when a line is regarded as a circle through infinity. Linear fractional transformations preserve the relation of symmetric points with respect to a circle or line, taking the reference circle or line to its image (the symmetry principle). The cross ratio is invariant under a linear fractional transformation. A linear fractional transformation is a composition of translations, rotations, magnifications, and the reciprocal map .

A linear fractional transformation is uniquely determined by the images of three distinct points in the extended complex plane, provided the three images are also distinct. Symmetry can supply a further image from a known pair of symmetric points and . If and the image of their reference circle or line is known, then , where is symmetric to with respect to that image. Substituting finite prescribed images in gives equations for the coefficients. Images involving infinity use the extension rules above. When , the coefficients can be scaled together to set . The case must be treated separately.