Mobius transformation :
Print this page
  
 
First   |   Last   |   Prev   |   Next
  


           When one of the points in either plane is the point at infinity, the quotient of the factors involving that            point in the above expression is to be replaced by $-1$ if that point is appearing after the minus sign,            otherwise to be replaced by $1$. For example, if $z_{2} = \infty$ then the factor $(z_{1} - z_{2})$ is to be replaced by $-1$            (since $z_{2}$ is appearing after the minus sign) and the factor $(z_{2} - z_{3})$ is to be replaced by $1$. Then, the            expression will become MATH and it will be solved for $w$.
           One can show that the Mobius transformation taking $z_{i}$ to $w_{i}$ where $z_{i}$ and $w_{i}$ are in $\QTR{Bbb}{C}$ can be expressed in            the determinant form as MATH       6 . If $z_{1}$, $z_{2}$, $z_{3}$ and $z_{4}$ are four distinct points in MATH, then the cross ratio of $z_{1}$, $z_{2}$, $z_{3}$ and $z_{4}$ is the image of $z_{1}$            under the unique Mobius transforation which takes $z_{2}$ to $0$, $z_{3}$ to $1$, and $z_{4}$ to $\infty$ and it is denoted by             MATH. The expression for the cross ratio MATH is given by MATH

 
First   |   Last   |   Prev   |   Next