  We generalize the definition of thin position of Scharlemann and Thompson for
compact orientable 3-manifolds with torus boundary components and introduce
$\alpha$-sloped generalized Heegaard splittings. We examine its relationship to
generalized Heegaard splittings of manifolds resulting from Dehn filling. We
compare alpha-sloped thin position of 3-manifolds to other types of thin
position for knots and 3-manifolds and discuss how this kind of decomposition
gives an organic picture of M and allows the structure of the manifold to
dictate the most natural slope(s) on the boundary. Additionally, we provide
illustrative examples and questions motivating the study of alpha-sloped thin
position.
