Osculating varieties and their joins: P1 x P1

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