Zametki na polyah (akor168) wrote,
Zametki na polyah
akor168

Matematical: fractional-linear transformations

Let's think about fractional-linear transformations. These are analytical functions on the complex plane C of the form f(z)=(az+b)/(cz+d) where a,b,c,d are complex, and ad-bc=1. This set of functions forms a group with respect to the composition which might be identified with SL(2,C) - special linear group of 2x2 matrices over C of determinant 1.

Each fractional linear transformation is actually is an automorphism of extended complex plane {C union with {infinity}} which via stereographic projection is nothing but S^2 - two dimensional sphere.

The following property of this group is remarkable:
given three points of C: z1,z2,z3 and other three points w1,w2,w3. Then there is a unique fractional -linear transformation f:C->C s.t. f(z1)=w1,f(z2)=w2,f(z3)=w3.
You can write this transformation explicitly:
((z-z1)(z3-z2))/((z-z2)(z3-z1)) =
((w-w1)(w3-w2))/((w-w2)(w3-w1))
and then solve for w=f(z).

Now the natural question is coming what if we have n points Z1,Z2,...Zn and W1,W2,...Wn.
Any examples of the families of analytical functions which form a group under the compositon such that we can find a unique function which transforms Z's to W's.

Of course, it would be nice to find as simple family as possible.
Tags: math
Subscribe

  • 3-0 vs 42-0

    To put the magnitude of the U.S. defeat in context, losing 3-0 in soccer is the equivalent of losing 42-0 in football. Реально улыбнуло, поскольку…

  • Анекдоты: полная потеря смысла при пересказе

    Знаете, когда обсуждается сложность перевода с одного языка на другой, обычно рассказывается пример с круглым столом где каждый знает языки двух…

  • полезность регулярных проф-заметок

    Терри Тао пишет аж в 2013 году(в комментах) про полезность ведения ЖЖ собственного блога, в котором можно записывать прочитанные результаты,…

  • Post a new comment

    Error

    default userpic

    Your reply will be screened

    Your IP address will be recorded 

    When you submit the form an invisible reCAPTCHA check will be performed.
    You must follow the Privacy Policy and Google Terms of use.
  • 0 comments