  We consider random flights in $\mathbb{R}^d$ reflecting on the surface of a
sphere $\mathbb{S}^{d-1}_R,$ with center at the origin and with radius $R,$
where reflection is performed by means of circular inversion. Random flights
studied in this paper are motions where the orientation of the deviations are
uniformly distributed on the unit-radius sphere $\mathbb{S}^{d-1}_1$.
  We obtain the explicit probability distributions of the position of the
moving particle when the number of changes of direction is fixed and equal to
$n\geq 1$. We show that these distributions involve functions which are
solutions of the Euler-Darboux-Poisson equation. The unconditional probability
distributions of the reflecting random flights are obtained by suitably
randomizing $n$ by means of a fractional-type Poisson process.
  Random flights reflecting on hyperplanes according to the optical reflection
form are considered and the related distributional properties derived.
