If $p,q$ are idempotents in a Banach algebra $A$ and
if $p+q-1$ is invertible, then the Kovarik formula provides an
idempotent $k(p,q)$ such that $pA=k(p,q)A$ and $Aq=Ak(p,q)$. We study
the existence of such an element in a more general situation. We first
show that $p+q-1$ is invertible if and only if $k(p,q)$ and $k(q,p)$
both exist. Then we deduce a local parametrization of the set of
idempotents from this equivalence. Finally, we consider a polynomial
parametrization first introduced by Holmes and we answer a
question raised at the end of his paper.