Minimal decomposition of binary forms with respect to tangential projections

Let $C\subset {\mathbb{P}}^n$ be a rational normal curve and let ${\mathrm{\ell }}_O:{\mathbb{P}}^{n+1}⇢{\mathbb{P}}^n$ be any tangential projection form a point $O\in T_{A}C$ where $A\in C$. Hence $X:= \ell_O©\subset $\mathbb{P}$^n$isalinearlynormalcuspidalcurvewithdegree$n+1$\mathrm{.}Forany$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 $B$.