  Recently we developed a diagonal homotopy method to compute a numerical
representation of all positive dimensional components in the intersection of
two irreducible algebraic sets. In this paper, we rewrite this diagonal
homotopy in intrinsic coordinates, which reduces the number of variables,
typically in half. This has the potential to save a significant amount of
computation, especially in the iterative solving portion of the homotopy path
tracker. There numerical experiments all show a speedup of about a factor two.
