Editing Möbius transformation (section)

=== 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.