  The paper is devoted to an approach to the notion of the complex dilatation
based on the following observations. (1) A natural measure of the distortion of
the conformal structure by a real linear automorphism of the complex plane is
the pull-back of the standard conformal structure of complex plane. (2) The set
of all conformal structures on the complex plane carries a canonical structure
of a model of the hyperbolic plane and can be naturally identified with the
unit disc together with its structure of the Klein model of the hyperbolic
plane. (3) The standard isomorphism of the Klein model with the Poincar\'e unit
disc model transforms this measure of distortion into the classical complex
dilatation.
  In version 2 this approach is related to Arnold's proof of the hyperbolic
altitudes theorem.
