  Using Connes distance formula in noncommutative geometry, it is possible to
retrieve the Euclidean distance from the canonical commutation relations of
quantum mechanics. In this note, we study modifications of the distance induced
by a deformation of the position-momentum commutation relations. We first
consider the deformation coming from a cut-off in momentum space, then the one
obtained by replacing the usual derivative on the real line with the h- and
q-derivatives, respectively. In these various examples, some points turn out to
be at infinite distance. We then show (on both the real line and the circle)
how to approximate points by extended distributions that remain at finite
distance. On the circle, this provides an explicit example of computation of
the Wasserstein distance.
