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Mar. 14th, 2003
Interesnye ssylki
Mar. 14th, 2003 07:16 pmГорчев о критиках
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О Русских сказках: нелогичность?
И еще о сказках - интересные комменты от eutrapelia
О "Паутине" Андреева
Воспоминания сельского ветеринара 7/10
Хорошо написано!(эпизод из студенческой жизни ветврача)
О корпоративной культуре: "семейный" вариант. Очень интересно!! 6/10
Записывайте сказки на трезвую голову!
О Русских сказках: нелогичность?
И еще о сказках - интересные комменты от eutrapelia
О "Паутине" Андреева
Воспоминания сельского ветеринара 7/10
Хорошо написано!(эпизод из студенческой жизни ветврача)
О корпоративной культуре: "семейный" вариант. Очень интересно!! 6/10
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.
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.