  The problem of determining finite subsets of characteristic planar model sets
(mathematical quasicrystals) $\varLambda$, called cyclotomic model sets, by
parallel $X$-rays is considered. Here, an $X$-ray in direction $u$ of a finite
subset of the plane gives the number of points in the set on each line parallel
to $u$. For practical reasons, only $X$-rays in $\varLambda$-directions, i.e.,
directions parallel to non-zero elements of the difference set $\varLambda -
\varLambda$, are permitted. In particular, by combining methods from algebraic
number theory and convexity, it is shown that the convex subsets of a
cyclotomic model set $\varLambda$, i.e., finite sets $C\subset \varLambda$
whose convex hulls contain no new points of $\varLambda$, are determined, among
all convex subsets of $\varLambda$, by their $X$-rays in four prescribed
$\varLambda$-directions, whereas any set of three $\varLambda$-directions does
not suffice for this purpose. We also study the interactive technique of
successive determination in the case of cyclotomic model sets, in which the
information from previous $X$-rays is used in deciding the direction for the
next $X$-ray. In particular, it is shown that the finite subsets of any
cyclotomic model set $\varLambda$ can be successively determined by two
$\varLambda$-directions. All results are illustrated by means of well-known
examples, i.e., the cyclotomic model sets associated with the square tiling,
the triangle tiling, the tiling of Ammann-Beenker, the T\"ubingen triangle
tiling and the shield tiling.
