  Let $C\subset \mathbb{P}^n$ be a rational normal curve and let
$\ell_O:\mathbb{P}^{n+1}\dashrightarrow \mathbb{P}^n$ be any tangential
projection form a point $O\in T_AC$ where $A\in C$. Hence $X:= \ell_O(C)\subset
\mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P
= \ell_O(B)$, $B\in \mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the
minimal cardinality of a set $S\subset X$ whose linear span contains $P$. Here
we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the
border $C$-rank of $B$.

Algebraic Geometry