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.
In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations.A special conformal transformation can be written
It is a composition of an inversion, a translation, and an inversion:
Its infinitesimal generator isThe inversion can also be taken to be multiplicative inversion of biquaternions B. The complex algebra B can be extended to P through the projective line over a ring. Homographies on P include translations:
The homography group G includes conjugates of translation by inversion:
The matrix describes the action of a special conformal transformation.
