proof of converse of Möbius transformation cross-ratio preservation theorem
Suppose that are distinct. Consider the transform defined as
Simple calculation reveals that , , and . Furthermore, equals the cross-ratio of .
Suppose we have two tetrads with a common cross-ratio . Then, as above, we may construct a transform which maps the first tetrad to and a transform which maps the first tetrad to . Then maps the former tetrad to the latter and, by the group property, it is also a Möbius transformation.
|Title||proof of converse of Möbius transformation cross-ratio preservation theorem|
|Date of creation||2013-03-22 17:01:51|
|Last modified on||2013-03-22 17:01:51|
|Last modified by||rspuzio (6075)|