This paper tackles the problem of globally computing sub-Riemannian curves on the Euclidean group of motions SE(3). In particular we derive a global result for special sub-Riemannian curves for which their Hamiltonian satisfies a particular condition. The sub-Riemannian curves in this paper are defined in the context of a constrained optimal control problem. The Maximun Principle is then applied to this problem to yield the appropriate left-invariant quadratic Hamiltonian. A number of integrable quadratic Hamiltonians are identified. We then proceed to derive convenient expressions for sub-Riemannian curves in SE(3) that correspond to particular extreme curves. These equations are then used to compute sub-Riemannian curves that could potentially be used for motion planning of underwater vehicles.