Editing Möbius transformation (section)

=== 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 [[conic]]s. The terminology is due to considering half the absolute value of the trace, |tr|/2, as the [[Eccentricity (mathematics)|eccentricity]] of the transformation&nbsp;– 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'' as a proxy for the eccentricity squared, as was done above; these classifications (but not the exact eccentricity values, since squaring and absolute values are different) agree for real traces but not complex traces. The same terminology is used for the [[SL2(R)#Classification of elements|classification of elements of {{nowrap|SL(2, '''R''')}}]] (the 2-fold cover), and [[Eccentricity (mathematics)#Analogous classifications|analogous classifications]] are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.