# Rotation of the Riemann Sphere

• Feb 16th 2009, 11:55 AM
srw899
Rotation of the Riemann Sphere
the mapping w=-1/z corresponds to a 180 degree rotation of the Riemann sphere about the x2-axis (the imaginary axis).
• Feb 16th 2009, 02:41 PM
ThePerfectHacker
Quote:

Originally Posted by srw899
the mapping w=-1/z corresponds to a 180 degree rotation of the Riemann sphere about the x2-axis (the imaginary axis).

The mapping $\displaystyle z\mapsto -\tfrac{1}{z}$ can be visualized as two mappings, the first $\displaystyle z\mapsto -z$ followed by $\displaystyle z\mapsto \tfrac{1}{z}$. Remember that $\displaystyle z\mapsto -z$ will rotate the Riemann sphere by 180 degrees while $\displaystyle z\to \tfrac{1}{z}$ will flip the Riemann sphere over upside down. Putting these effects together we get exactly what $\displaystyle z\mapsto -\tfrac{1}{z}$ does.
• Feb 16th 2009, 02:53 PM
srw899
another question
show The mapping w=(e^i*theta)z corresponds to a rotation of the riemann sphere about the xyaxis through an angle theta.