Post by Admin on Oct 26, 2016 8:05:22 GMT
Gary
As I was re-reading the stuff about Möbius transformations I thought I might as well post my solution to this suggested exercise.
In the LHS of figure 33 on page 168 we have two families of circles. All the circles pass through the point $\xi$ and each family is orthogonal to the other at $\xi$ and at their second intersection points. The Möbius transformation $G$ sends the point $\xi$ to $\infty$, and since Möbius transformations send circles to circles, each family of circles will therefore be sent to a family of parallel lines. Since Möbius transformations are also conformal, the two families of parallel lines will be orthogonal, as shown in the RHS of figure 33.
Vasco
As I was re-reading the stuff about Möbius transformations I thought I might as well post my solution to this suggested exercise.
In the LHS of figure 33 on page 168 we have two families of circles. All the circles pass through the point $\xi$ and each family is orthogonal to the other at $\xi$ and at their second intersection points. The Möbius transformation $G$ sends the point $\xi$ to $\infty$, and since Möbius transformations send circles to circles, each family of circles will therefore be sent to a family of parallel lines. Since Möbius transformations are also conformal, the two families of parallel lines will be orthogonal, as shown in the RHS of figure 33.
Vasco
