  Guided by the Hopf fibration, we single out a family (indexed by a positive
constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using
the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a
privileged right invariant Killing field of constant length. Each such
Riemannian metric pairs with the corresponding Killing field to produce a
y-global and {\it explicit} Randers metric on $S^3$. Using the machinery of
spray curvature and Berwald's formula for it, we prove directly that the said
Randers metric has constant positive flag curvature K, as predicted by the
Yasuda-Shimada theorem. We also explain why this family of Finslerian space
forms is NOT projectively flat.
