Editing Möbius transformation (section)

=== Elliptic transformations ===
If {{nowrap|1=''ρ'' = 0}}, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be ''elliptic''. These transformations tend to move all points in circles around the two fixed points. If one of the fixed points is at infinity, this is equivalent to doing an affine rotation around a point.

If we take the [[one-parameter subgroup]] generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere. In general, the two fixed points can be any two distinct points.

This has an important physical interpretation.
Imagine that some observer rotates with constant angular velocity about some axis. Then we can take the two fixed points to be the North and South poles of the celestial sphere. The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, ∞, and with the number ''α'' corresponding to the constant angular velocity of our observer.

Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):

[[File:Mobius Small Neg Elliptical.jpeg]]

[[File:Mobius Large Pos Elliptical.jpeg]]

These pictures illustrate the effect of a single Möbius transformation. The one-parameter subgroup which it generates ''continuously'' moves points along the family of circular arcs suggested by the pictures.