si is the reflection with respect to the hyperplane defined by αi = 0, because it preserves the scalar product, fixes the plane orthogonal to αi and maps αi on − αi. 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 αi(β ), i = 1,2,3, really represent the values of the linear maps αi : β → αi(β) ∈ ℝ. The billiard ball moves in the space of scale factors, say ℳ β (β-space), and hence the maps αi, which define the walls, belong to the dual space ⋆ ℳ β of linear forms acting on ℳ β. In order to be compatible with the treatment in Section 2.4 (cf. Equation (2.45View Equation)), Equation (3.7View Equation) – even though written here as a reflection in the space ⋆ ℳ β – really corresponds to a geometric reflection in the space ℳ β, in which the particle moves. This will be carefully explained in Section 5.2 (cf. Equations (5.20View Equation) and (5.21View Equation)), after the necessary mathematical background has been introduced.