  We continue the study of intersection bodies of polytopes, focusing on the
behavior of $IP$ under translations of $P$. We introduce an affine hyperplane
arrangement and show that the polynomials describing the boundary of $I(P+t)$
can be extended to polynomials in variables $t\in \mathbb{R}^d$ within each
region of the arrangement. In dimension $2$, we give a full characterization of
those polygons such that their intersection body is convex. We give a partial
characterization for general dimensions.
