Reconstruction of a submanifold of Euclidean space from its Grassmannian image that degenerates into a line

We study the existence of a submanifold $F^n$ of Euclidean space $E^{n+p}$ with prescribed Grassmannian image that degenerates into a line. We prove that $\Gamma$ is the Grassmannian image of a regular submanifold $F^n$ of Euclidean space $E^{n+p}$ if and only if the curve $\Gamma$ in the Grassmann manifold $G^+(p,n+p)$ is asymptotically $C^r$-regular, $r>1$. Here $G^+(p,n+p)$ is embedded into the sphere $S^N$, $N=C_{n+p}^p$, by the Plücker coordinates.