projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

, a special conformal transformation is a

linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transf ...

that is ''not'' an

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...

. Thus the

generation A generation refers to all of the people born and living at about the same time, regarded collectively. It can also be described as, "the average period, generally considered to be about 20–⁠30 years, during which children are born and gr ...

of a special conformal transformation involves use of multiplicative inversion, which is the generator of linear fractional transformations that is not affine. In

mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

, certain

conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...

s known as

spherical wave transformation Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman givi ...

s are special conformal transformations.

Vector presentation

A special conformal transformation can be written :

x'^\mu = \frac = \frac(x^\mu-b^\mu x^2)\,.

It is a composition of an

inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...

(''x''^''μ'' → ''x''^''μ''/x² = ''y''^''μ''), a

translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...

(''y''^''μ'' → ''y''^''μ'' − ''b''^''μ'' = ''z''^''μ''), and another inversion (''z''^''μ'' → ''z''^''μ''/z² = ''x''′^''μ'') :

\frac = \frac - b^\mu \,.

Its infinitesimal generator is :

K_\mu = -i(2x_\mu x^\nu\partial_\nu - x^2\partial_\mu) \,.

Alternative presentation

The inversion can also be taken Arthur Conway (1911) "On the application of quaternions to some recent developments of electrical theory", ''Proceedings of the Royal Irish Academy'' 29:1–9, particularly page 9 to be multiplicative inversion of

biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...

s ''B''. The complex algebra ''B'' can be extended to P(''B'') through the

projective line over a ring In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring ''A'' with 1, the projective line P(''A'') over ''A'' consists of points identified by projective coordinates. Let ''U'' ...

. Homographies on P(''B'') include translations: :

U(q,1) \begin1 & 0 \\ t & 1 \end = U(q + t, 1).

The homography group G(''B'') includes conjugates of translation by inversion: :

\begin0 & 1 \\ 1 & 0 \end\begin1 & 0 \\ t & 1\end\begin0 & 1 \\ 1 & 0 \end = \begin1 & t \\ 0 & 1\end.

The matrix describes the action of a special conformal transformation.

References

{{Reflist Projective geometry Conformal field theory