is the reflection with respect to the hyperplane defined by
, because it preserves the scalar product, fixes the
plane orthogonal to
and maps
on
. Note that we are here being deliberately careless about notation in order
not to obscure the main point, namely that the billiard reflections are elements of a Coxeter group. To be precise, the linear
forms
,
really represent the values of the linear maps
. The billiard ball moves in
the space of scale factors, say
(
-space), and hence the maps
, which define the walls, belong to the
dual space
of linear forms acting on
. In order to be compatible with the treatment in Section 2.4
(cf. Equation (2.45)), Equation (3.7) – even though written here as a reflection in the space
– really
corresponds to a geometric reflection in the space
, in which the particle moves. This will be carefully
explained in Section 5.2 (cf. Equations (5.20) and (5.21)), after the necessary mathematical background has been
introduced.