and only if |λ| = 1 and λ ≠ ±1. Writing , an elliptic transform is conjugate to with α real. Note that for any with characteristic constant k, the characteristic constant of is kn. Thus, all Möbius transformations of finite order are elliptic transformations, namely exactly those where λ is a root of unity, or, equivalently, where α is a rational multiple of . The simplest possibility of a fractional multiple means α = /2, which is also the unique case of , is also denoted as a ; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as: There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: which fixes 1 and swaps 0 with ∞ (rotation by 180° about the points 1 and −1), , which fixes ∞ and swaps 0 with 1 (rotation by 180° about the points 1/2 and ∞), and which fixes 0 and swaps 1 with ∞ (rotation by 180° about the points 0 and 2). Hyperbolic transforms The transform is said to be hyperbolic if it can be represented by a matrix whose trace is real with A transform is hyperbolic if and only if λ is real and λ ≠ ±1. Loxodromic transforms The transform is said to be loxodromic if is not in [0,4]. A transformation is loxodromic if and only if . Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below. General classification The real case and a note on terminology Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real conics. The terminology is due to considering half the absolute value of the trace, |tr|/2, as the eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/n is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. Alternatively one may use half the trace squared