 1.  The M�bius transformations are the projective transformations of the complex projective line.

 2.  The crossratio is invariant under the projective transformations of the line.

 3.  These include both affine transformations ( such as translation ) and projective transformations.

 4.  The restriction of this projective transformation to the midsphere is a M�bius transformation.

 5.  We see that projective transformations don't mix Gaussian curvatures of different sign.

 6.  It is the study of geometric properties that are invariant with respect to projective transformations.

 7.  Two matrices represent the same projective transformation if one is a constant multiple of the other.

 8.  Projective transformations are represented by matrices.

 9.  :Crossratio is a projective invariant i . e . it is unchanged by projective transformations.

 10.  But I'm working with projective transformations now, and these volume elements aren't preserved.
