Osculating varieties and their joins: $\mathbb{P}^1\times \mathbb{P}^1$

Let $X\subset \PP^r$ be an integral projective variety. We study the dimensions of joins several copies osculating varieties $J(X,m)$ $X$. Our methods are general, but we give a full description in all cases only if $X$ is linearly normal embedding $\PP^1\times \PP^1$. For these embeddings \PP^1$ examples and then one copy arbitrary number