  The quantum state space $\cal S$ over a $d$-dimensional Hilbert space is
represented as a convex subset of a $D-1$-dimensional sphere $S_{D-1}\subset
{\bf{R}}^D$, where $D=d^2-1.$ Quantum tranformations (CP-maps) are then
associated with the affine transformations of ${\bf{R}}^D,$ and $N\mapsto M$
{\it cloners} induce polynomial mappings. In this geometrical setting it is
shown that an optimal cloner can be chosen covariant and induces a map between
reduced density matrices given by a simple contraction of the associated
$D$-dimensional Bloch vectors.
