Linear fractional transformation

In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers.

More generally in mathematics, C may be replaced by another ring (A, +, ×).^[1] For example, the Cayley transform is a linear fractional transformation originally defined on the 3 x 3 real matrix ring.

In general, a linear fractional transformation refers to a homography of P(A), the projective line over a ring A. When A is a commutative ring, then a linear fractional transformation has the familiar form

where a, b, c, d are elements of A such that ac – bd is a unit of a (that is ac – bd has a multiplicative inverse in A)

In a non-commutative ring (A, +, ×), with (z,t) in A², the units u determine an equivalence relation An equivalence class in the projective line over A is written U(z,t). Then linear fractional transformations act on the right of an element of P(A):

The ring is embedded in its projective line by z → U(z,1), so t = 1 recovers the usual expression. This linear fractional transformation is well-defined since U(za + tb, zc + td) does not depend on which element is selected from its equivalence class for the operation.

The linear fractional transformations form a group, denoted

The group of the linear fractional transformations is called the modular group. It has been widely studied because its numerous applications to number theory, which include, in particular, Wiles's proof of Fermat's Last Theorem.

Conformal property

The commutative rings of split-complex numbers and dual numbers join the ordinary complex numbers as rings that express angle. In each case the exponential map applied to the imaginary axis produces an isomorphism between one-parameter groups in (A, + ) and in the group of units (U, × ):

The "angle" y is hyperbolic angle, slope, or circular angle according to the host ring.

A linear fractional transformation can be generated by multiplicative inversion z → 1/z and affine transformations z → a z + b. Conformality can be confirmed by showing the generators are all conformal. The translation z → z + b is a change of origin and makes no difference to angle. To see that z → az is conformal, consider the polar decomposition of a and z. In each case the angle of a is added to that of z giving a conformal map. Finally, inversion is conformal since z → 1/z sends

References

• B.A. Dubrovin, A.T. Fomenko, S.P. Novikov (1984) Modern Geometry — Methods and Applications, volume 1, chapter 2, §15 Conformal transformations of Euclidean and Pseudo-Euclidean spaces of several dimensions, Springer-Verlag ISBN 0-387-90872-2.
• Geoffry Fox (1949) Elementary Theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane, Master’s thesis, University of British Columbia.
• P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A 51:67–85.
• A.E. Motter & M.A.F. Rosa (1998) "Hyperbolic calculus", Advances in Applied Clifford Algebras 8(1):109 to 28, §4 Conformal transformations, page 119.
• Tsurusaburo Takasu (1941) Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2, Proceedings of the Imperial Academy 17(8): 330–8, link from Project Euclid, MR 14282
• Isaak Yaglom (1968) Complex Numbers in Geometry, page 130 & 157, Academic Press