which sends the points (γ1, γ2) to (0, ∞). Here we assume that γ1 and γ2 are distinct and finite. If one of them is already at infinity then g can be modified so as to fix infinity and send the other point to 0. If f has distinct fixed points (γ1, γ2) then the transformation has fixed points at 0 and ∞ and is therefore a dilation: . The fixed point equation for the transformation f can then be written Solving for f gives (in matrix form): or, if one of the fixed points is at infinity: From the above expressions one can calculate the derivatives of f at the fixed points: and Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (k) of f as the characteristic constant of f. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant: For loxodromic transformations, whenever |k| > 1, one says that γ1 is the repulsive fixed point, and γ2 is the attractive fixed point. For |k| < 1, the roles are reversed. Parabolic case: In the parabolic case there is only one fixed point γ. The transformation sending that point to ∞ is or the identity if γ is already at infinity. The transformation fixes infinity and is therefore a translation: Here, β is called the translation length. The fixed point formula for a parabolic transformation is then Solving for f (in matrix form) gives or, if γ = ∞: Note that β is not the characteristic constant of f, which is always 1 for a parabolic transformation. From the above expressions one can calculate: Poles of the transformation The point is called the pole of ; it is that point which is transformed to the point at infinity under . The inverse pole is that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points: These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation. A transform can be specified with two fixed points γ1, γ2 and the pole . This allows us to derive a formula for conversion between k and given : which reduces down to The last expression coincides with one of