Editing Möbius transformation (section)

=== Hyperbolic transformations ===

If ''α'' is zero (or a multiple of 2{{pi}}), then the transformation is said to be ''hyperbolic''. These transformations tend to move points along circular paths from one fixed point toward the other.

If we take the [[one-parameter group|one-parameter subgroup]] generated by any hyperbolic 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 certain family of circular arcs ''away'' from the first fixed point and ''toward'' the second fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.

This too has an important physical interpretation. Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points 0, ∞, with the real number ''ρ'' corresponding to the magnitude of his acceleration vector. The stars seem to move along longitudes, away from the South pole toward the North pole. (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane.)

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

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

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

These pictures resemble the field lines of a positive and a negative electrical charge located at the fixed points, because the circular flow lines subtend a constant angle between the two fixed points.