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 special conformal transformation can be written
Its infinitesimal generator is
The inversion can also be taken 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.
- Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. pp. 97–98. ISBN 978-0-387-94785-3.
- 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