Special conformal transformation

Projective presentation

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:

${\displaystyle U(q:1)\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right)=U(q+t:1).}$

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

${\displaystyle \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right).}$

The matrix describes the action of a special conformal transformation.

Group property

The translations form a subgroup of the linear fractional group acting on a projective line. There are two embeddings into the projective line of homogeneous coordinates: z → [z:1] and z → [1:z]. An addition operation corresponds to a translation in the first embedding. The translations to the second embedding are special conformal transformations, forming translations at infinity. Addition by these transformations reciprocates the terms before addition, then returns the result by another reciprocation. This operation is called the parallel operation. In the case of the complex plane the parallel operator forms an addition operation in an alternative field using infinity but excluding zero. The translations at infinity thus form another subgroup of the homography group on the projective line.

History

The term special conformal transformation ("speziellen konformen Transformationen" in German) was first used in 1962 by Hans Kastrup.