  From physics standpoint exotic matter problem is a major difficulty in
thin-shell wormholes (TSWs) with spherical / cylindrical throat topologies. We
aim to circumvent this handicap by considering angular dependent throats in
$3+1-$dimensions. By considering the throat of the TSW to be deformed
spherical, i.e., a function of $\theta $ and $\varphi $, we present general
conditions which are to be satisfied by the shape of the throat in order to
have the wormhole supported by matter with positive density in the static
reference frame. We provide particular solutions / examples to the constraint
conditions.
