moves points along the family of circular arcs suggested by the pictures. Hyperbolic transformations If α is zero (or a multiple of 2), 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 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): 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 – 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