  Neutral meson mixing and $CP$ violation are very well known weak processes
that involve decays to meson states that are, in general, a superposition of
flavor eigenstates. This paper describes a mathematical interpretation of the
time-dependent mixing amplitudes as a complex hyperbolic rotation of the time
evolution of those amplitudes without mixing, which involves a Lie group
$SO(1,1,\mathbb{C})$.
  This allows a geometric interpretation of mixing as a curve into the
$SO(1,1,\mathbb{C})$ manifold, parameterized with the proper decay time, where
$CP$ violation is the image of this curve at $t = 0$.
  To show the power of this new interpretation, it is applied to several
aspects of the measurement of the CKM angle $\gamma$ in $B$ decays to neutral
$D$ mesons. On one hand, the charm mixing correction on the $CPV$ parameters is
derived. On the other hand, it is shown how the expressions used in GLW, ADS
and GGSZ methods are affected by charm mixing. Finally, the complete example
with both charm and strange mixing and $CPV$ is described.
