In this paper we improve the known bound for the $X$-rank $R_{ X}(P)$ of an element $P\in { \mathbb{ P}}^N$ in the case in which $X\subset { \mathbb P}^n$ is a projective variety obtained as a linear projection from a general $v$-dimensional subspace $V\subset { \mathbb P}^{ n+v}$. Then, if $X\subset { \mathbb P}^n$ is a curve obtained from a projection of a rational normal curve $C\subset { \mathbb P}^{ n+1}$ from a point $O\subset { \mathbb P}^{ n+1}$, we are able to describe the precise value of the $X$-rank for those points $P\in { \mathbb P}^n$ such that $R_{ X}(P)\leq R_{ C}(O)-1$ and to improve the general result. Moreover we give a stratification, via the $X$-rank, of the osculating spaces to projective cuspidal projective curves $X$. Finally we give a description and a new bound of the $X$-rank of subspaces both in the general case and with respect to integral non-degenerate projective curves.
Categories: math.AG math.AC