  Let $X\subset \mathbb {P}^n$ be a linearly normal elliptic curve. For any
$P\in \mathbb {P}^n$ the $X$-rank of $P$ is the minimal cardinality of a set
$S\subset X$ such that $P\in \langle S\rangle$. In this paper we give an almost
complete description of the stratification of $\mathbb {P}^n$ given by the
$X$-rank and the open $X$-rank.
