special conformal transformation

{{Short description|Special class of linear fractional transformations}}

Image:Conformal grid before Möbius transformation.svg

Image:Conformal grid after Möbius transformation.svg

In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation. Thus the generation 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, certain conformal maps known as spherical wave transformations are special conformal transformations.

==Vector presentation==

A special conformal transformation can be written{{cite book |last = Di Francesco |author2=Mathieu, Sénéchal |title= Conformal field theory |series= Graduate texts in contemporary physics |year= 1997 |publisher= Springer |isbn= 978-0-387-94785-3 |pages= 97–98}}

:$x\text{'}^\backslash mu\; =\; \backslash frac\{x^\backslash mu-b^\backslash mu\; x^2\}\{1-2b\backslash cdot\; x+b^2x^2\}\; =\; \backslash frac\{x^2\}\{|x-bx^2|^2\}(x^\backslash mu-b^\backslash mu\; x^2)\backslash ,.$

It is a composition of an inversion (x^μ → x^μ/x² = y^μ), a translation (y^μ → y^μ − b^μ = z^μ), and another inversion (z^μ → z^μ/z² = x′^μ)

:$\backslash frac\{x\text{'}^\backslash mu\}\{x\text{'}^2\}\; =\; \backslash frac\{x^\backslash mu\}\{x^2\}\; -\; b^\backslash mu\; \backslash ,.$

Its infinitesimal generator is

:$K\_\backslash mu\; =\; -i(2x\_\backslash mu\; x^\backslash nu\backslash partial\_\backslash nu\; -\; x^2\backslash partial\_\backslash mu)\; \backslash ,.$

Special conformal transformations have been used to study the force field of an electric charge in hyperbolic motion.Galeriu, Cǎlin (2019) "Electric charge in hyperbolic motion: the special conformal solution", European Journal of Physics 40(6) {{doi| 10.1088/1361-6404/ab3df6}}

==Projective presentation==

The inversion can also be takenArthur 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 biquaternions B. The complex algebra B can be extended to P(B) through the projective line over a ring. Homographies on P(B) include translations:

:

The homography group G(B) includes of translations at infinity with respect to the embedding q → U(q:1);

:

The matrix describes the action of a special conformal transformation.{{Wikibooks-inline|Associative Composition Algebra/Homographies}}