Editing Möbius transformation (section)

== Geometric interpretation of the characteristic constant ==
The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case:

[[File:Mobius Identity.jpeg]]

The characteristic constant can be expressed in terms of its [[Natural logarithm|logarithm]]:
<math display="block">e^{\rho + \alpha i} = k. </math>
When expressed in this way, the real number ''ρ'' becomes an expansion factor. It indicates how repulsive the fixed point ''γ''<sub>1</sub> is, and how attractive ''γ''<sub>2</sub> is. The real number ''α'' is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about ''γ''<sub>1</sub> and clockwise about&nbsp;''γ''<sub>2</sub>.

=== 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.

=== 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.

=== Loxodromic transformations ===

If both ''ρ'' and ''α'' are nonzero, then the transformation is said to be ''loxodromic''. These transformations tend to move all points in S-shaped paths from one fixed point to the other.

The word "[[loxodrome]]" is from the Greek: "λοξος (loxos), ''slanting'' + δρόμος (dromos), ''course''". When [[sailing]] on a constant [[bearing (navigation)|bearing]] – if you maintain a heading of (say) north-east, you will eventually wind up sailing around the [[north pole]] in a [[logarithmic spiral]]. On the [[mercator projection]] such a course is a straight line, as the north and south poles project to infinity. The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of ''k''. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes.

If we take the [[one-parameter group|one-parameter subgroup]] generated by any loxodromic 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 curves, ''away'' from the first fixed point and ''toward'' the second fixed point. Unlike the hyperbolic case, these curves are not circular arcs, but certain curves which under stereographic projection from the sphere to the plane appear as spiral curves which twist counterclockwise infinitely often around one fixed point and twist clockwise infinitely often around the other fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.

You can probably guess the physical interpretation in the case when the two fixed points are 0,&nbsp;∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the ''same'' axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ''ρ'', ''α'' determined respectively by the magnitude of the actual linear and angular velocities.

=== Stereographic projection ===
These images show Möbius transformations [[stereographic projection|stereographically projected]] onto the [[Riemann sphere]]. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location.
{|
|-
| colspan="3" style="text-align:center;"| '''One fixed point at infinity'''
|-
| style="text-align:center;"| [[File:Mob3d-elip-inf-480.png|thumb|Elliptic]]
| style="text-align:center;"| [[File:Mob3d-hyp-inf-480.png|thumb|Hyperbolic]]
| style="text-align:center;"| [[File:Mob3d-lox-inf-480.png|thumb|Loxodromic]]
|-
| colspan="3" style="text-align:center;"| '''Fixed points diametrically opposite'''
|-
| style="text-align:center;"| [[File:Mob3d-elip-opp-480.png|thumb|Elliptic]]
| style="text-align:center;"| [[File:Mob3d-hyp-opp-480.png|thumb|Hyperbolic]]
| style="text-align:center;"| [[File:Mob3d-lox-opp-480.png|thumb|Loxodromic]]
|-
| colspan="3" style="text-align:center;"| '''Fixed points in an arbitrary location'''
|-
| style="text-align:center;"| [[File:Mob3d-elip-arb-480.png|thumb|Elliptic]]
| style="text-align:center;"| [[File:Mob3d-hyp-arb-480.png|thumb|Hyperbolic]]
| style="text-align:center;"| [[File:Mob3d-lox-arb-480.png|thumb|Loxodromic]]
|}