  Let $P$ be a set of $n$ labeled points in the plane. The radial system of $P$
describes, for each $p\in P$, the order in which a ray that rotates around $p$
encounters the points in $P \setminus \{p\}$. This notion is related to the
order type of $P$, which describes the orientation (clockwise or
counterclockwise) of every ordered triple in $P$. Given only the order type,
the radial system is uniquely determined and can easily be obtained. The
converse, however, is not true. Indeed, let $R$ be the radial system of $P$,
and let $T(R)$ be the set of all order types with radial system $R$ (we define
$T(R) = \emptyset$ for the case that $R$ is not a valid radial system).
Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings,
in ISAAC 2014) show that $T(R)$ may contain up to $n-1$ order types. They also
provide polynomial-time algorithms to compute $T(R)$ when only $R$ is given.
  We describe a new algorithm for finding $T(R)$. The algorithm constructs the
convex hulls of all possible point sets with the radial system $R$. After that,
orientation queries on point triples can be answered in constant time. A
representation of this set of convex hulls can be found in $O(n)$ queries to
the radial system, using $O(n)$ additional processing time. This is optimal.
Our results also generalize to abstract order types.
