In recent years, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth-Moon system and in alternative multibody environments, and several space missions have already taken advantage of the results of the related studies. Recent efforts have been devoted to developing a suitable representation for the manifolds, which would be extremely useful for mission analysis and optimization. This work describes an intuitive polyhedral interpolative approach for each state component associated with manifold trajectories, both in two and in three dimensions. A grid of data, coming from the numerical propagation of a finite number of manifold trajectories, is used. This representation is employed for some invariant manifolds, associated with the two planar Lyapunov orbits at the collinear libration points located in the proximity of the Moon, and with a three-dimensional Halo orbit. Accuracy is evaluated, and is proven to be satisfactory, with the exclusion of limited regions of the manifolds. The polyhedral interpolation technique has several applications. Four of them are illustrated in detail in this paper. First, orbits transiting from the Earth to the Moon are identified in the phase space. Second, a variety of homoclinic trajectories connected with two Lyapunov orbits are detected. Third, the polyhedral interpolating technique is successfully employed for the determination of heteroclinic connections between the manifolds associated with the Lyapunov orbits. Lastly, near-homoclinic connections between the manifolds emanating from a Halo orbit are detected. These applications unequivocally prove utility and effectiveness of the polyhedral interpolative technique, and represent the premise for its application to space mission analysis involving invariant manifold dynamics.