The Shape Lemma was originally introduced in [3] and so christened by Lakshman ([5]). It is an easy generalization of the Primitive Element Theorem and it states that a Odimensional radical ideal in a polynomial ring k[X1, . . . . Xn], after most changes of coordinates, has a basis {91(X1),X2 -92(X1 ),. ~.,xn -gn(xl)} Notwithstanding its triviality, it has proved ubiquitous in recent papers on polynomial system solving ([1, 2, 4, 6, 7]). The obvious example (X2, XY, Y2) is sufficient to show that some assumption is needed on a O-dimensional ideal in order that it holds; the obvious example (X2, Y) is sufficient to show that radicality is too strong an assumption. Since most of the results making use of the Shape Lemma are valid whenever the Shape Lemma holds and are of interest also for non radical ideals, it is worthwhile to exactly characterize those O-dimensional ideals to which the Shape Lemma applies. It turns out that this exact characterization is as trivial as the original Shape Lemma itself. In fact both this characterization and the generalization of it we give are easy specializations of a classical result in algebraic geometry on the minimum dimension of a generic biregular projection of a variety as a function of its dimension and of the dimension of its tangent bundle. We give a direct, elementary, self-contained proof of this specialization.