  A theory of transversely oriented spun-normal immersed surfaces in ideally
triangulated 3--manifolds is developed in this paper, including linear
functionals determining the boundary curves, Euler characteristic and homology
class of these immersions. This is used to develop and implement an algorithm
to compute the unit ball of the Thurston norm for cusped hyperbolic
3--manifolds of finite volume. As an application of independent interest, we
give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces
with number of cusps bounded linearly in genus.
