  We use the vector wedge product in geometric algebra to show that Poisson
commutator brackets measure preservation of phase space areas. We also use the
vector dot product to define the Poisson anticommutator bracket that measures
the preservation of phase space angles. We apply these brackets to the paraxial
meridional complex height-angle ray vectors that transform via a 2x2 matrix,
and we show that this transformation preserves areas but not angles in phase
space. The Poisson brackets here are expressed in terms of the coefficients of
the ABCD matrix. We also apply these brackets to the distance-height ray
vectors measured from the input and output sides of the optical system. We show
that these vectors obey a partial Moebius transformation, and that this
transformation preserves neither areas nor angles. The Poisson brackets here
are expressed in terms of the transverse and longitudinal magnifications.
