  In this paper we consider quintessence reconstruction of interacting
holographic dark energy in a non-flat background. As system's IR cutoff we
choose the radius of the event horizon measured on the sphere of the horizon,
defined as $L=ar(t)$. To this end we construct a quintessence model by a real,
single scalar field. Evolution of the potential, $V(\phi)$, as well as the
dynamics of the scalar field, $\phi$, are obtained according to the respective
holographic dark energy. The reconstructed potentials show a cosmological
constant behavior for the present time. We constrain the model parameters in a
flat universe by using the observational data, and applying the Monte Carlo
Markov chain simulation. We obtain the best fit values of the holographic dark
energy model and the interacting parameters as
$c=1.0576^{+0.3010+0.3052}_{-0.6632-0.6632}$ and
$\zeta=0.2433^{+0.6373+0.6373}_{-0.2251-0.2251}$, respectively. From the data
fitting results we also find that the model can cross the phantom line in the
present universe where the best fit value of of the dark energy equation of
state is $w_D=-1.2429$.
