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 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, ∞: 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 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. Iterating a transformation If a transformation has fixed points γ1, γ2, and characteristic constant k, then will have . This can be used to iterate a transformation, or to animate one by breaking it up into steps. These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants. And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. Note that in the elliptical and loxodromic images, the α value is 1/10 . Higher dimensions In higher dimensions, a Möbius transformation is a homeomorphism of , the one-point compactification of , which is a finite composition of inversions in spheres and reflections in hyperplanes. Liouville's theorem in conformal geometry states that in dimension at least three, all conformal transformations are Möbius transformations. Every Möbius transformation can be put in the