  Given $r>n$ general hyperplanes in $\mathbb P^n,$ a star configuration of
points is the set of all the $n$-wise intersection of them. We introduce {\it
contact star configurations}, which are star configurations where all the
hyperplanes are osculating to the same rational normal curve. In this paper we
find a relation between this construction and Hadamard products of linear
varieties. Moreover, we study the union of contact star configurations on a
same conic in $\mathbb P^2$, we prove that the union of two contact star
configurations has a special $h$-vector and, in some cases, this is a complete
intersection.
