There are homographies that take
to some permutation of , where may differ
from . If is one of the homographies
that map to , then satisfies
(31.2.1) if is a solution of
(31.2.1) with replaced by and appropriately
transformed parameters. For example, if , then the parameters
are , ; ,
. If is one of the
homographies that do not map to , then an
appropriate prefactor must be included on the right-hand side. For example,
, which arises from
, satisfies (31.2.1) if
is a solution of (31.2.1) with replaced by and
transformed parameters ,
;
, .