  Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a
subfield of $L$.
  Consider $L$ as a vector space over $F$ and the associated projective space
that is isomorphic to ${\mathrm{PG}}(n-1,q)$.
  The properties of the projective mapping induced by $x\mapsto x^{-1}$ have
been studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that the
image of any line is a normal rational curve in some subspace.
  In this note a more detailed geometric description is achieved.
  Consequences are found related to mixed partitions of the projective spaces;
in particular, it is proved that for any positive integer $k$, if $q\ge2^k-1$,
then there are partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves
of degree $2^k-1$.
  For smaller $q$ the same construction gives partitions in $(q+1)$-tuples of
independent points.
