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, +, ×). 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
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.
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, × ):
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
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