  A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analytic
foliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leaves
other than straight lines have no inflection points. The classification of the
convex foliations of degree $2$ on $\mathbb P^2_{\mathbb C}$ has been
established in $2015$ by C.~\textsc{Favre} and J.~\textsc{Pereira}. The main
argument of this classification was a result obtained in~$2004$
by~D.~\textsc{Schlomiuk} and N.~\textsc{Vulpe} concerning the real polynomial
vector fields of degree $2$ whose associated foliation on
$\mathbb{P}^{2}_{\mathbb{R}}$ is convex. We present here a new proof of this
classification, that is simpler, does not use this result and does not leave
the holomorphic framework. It is based on the properties of certain models of
convex foliations of $\mathbb P^2_{\mathbb C}$ of arbitrary degree and of the
discriminant of the dual web of a foliation of $\mathbb P^2_{\mathbb C}$.
