Special conformal transformation

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.

A coordinate grid prior to a special conformal transformation
The same grid after a special conformal transformation

In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations.

Vector presentation

A special conformal transformation can be written[1]

It is a composition of an inversion (xμ  xμ/x2 = yμ), a translation (yμ  yμ  bμ = zμ), and another inversion (zμ  zμ/z2 = x′ μ) such that:

Its infinitesimal generator is

Alternative presentation

The inversion can also be taken[2] 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 conjugates of translation by inversion:

The matrix describes the action of a special conformal transformation.

References

  1. Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. pp. 97–98. ISBN 978-0-387-94785-3.
  2. 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
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