  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.
