  In the context of $f(\mathcal{R})$ gravity, dark energy is a geometrical
fluid with negative equation of state. Since the function $f(\mathcal{R})$ is
not known \emph{a priori}, the need of a model independent reconstruction of
its shape represents a relevant technique to determine which $f(\mathcal{R})$
model is really favored with respect to others. To this aim, we relate
cosmography to a generic $f(\mathcal R)$ and its derivatives in order to
provide a model independent investigation at redshift $z \sim 0$. Our analysis
is based on the use of three different cosmological distance definitions, in
order to alleviate the duality problem, i.e. the problem of which cosmological
distance to use with specific cosmic data sets. We therefore consider the
luminosity, $d_L$, flux, $d_F$, and angular, $d_A$, distances and we find
numerical constraints by the Union 2.1 supernovae compilation and measurement
of baryonic acoustic oscillations, at $z_{BAO}=0.35$. We notice that all
distances reduce to the same expression, i.e. $d_{L;F;A}\sim\frac{1}{\mathcal
H_0}z$, at first order. Thus, to fix the cosmographic series of observables, we
impose the initial value of $H_0$ by fitting $\mathcal H_0$ through supernovae
only, in the redshift regime $z<0.4$. We find that the pressure of curvature
dark energy fluid is slightly lower than the one related to the cosmological
constant. This indicates that a possible evolving curvature dark energy
realistically fills the current universe. Moreover, the combined use of
$d_L,d_F$ and $d_A$ shows that the sign of the acceleration parameter agrees
with theoretical bounds, while its variation, namely the jerk parameter, is
compatible with $j_0>1$. Finally, we infer the functional form of
$f(\mathcal{R})$ by means of a truncated polynomial approximation, in terms of
fourth order scale factor $a(t)$.
