Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to contain a positive definite matrix (the quadratic hypersurfaces of $A$ and $B$ have trivial intersection).

Has work been done on characterizing the cases when $P(A,B)$ contains a copositive matrix?

UPDATE:
Uhlig's paper *A recurring theorem about pairs of quadratic forms and extensions: a survey* is a convenient reference.