Rational normal curves and Hadamard products

Given r > n general hyperplanes in Pn; a star configuration of points is the set of all the n-wise intersection of them. We introduce 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 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.