Recent progress in diffusion imaging has lead to in-vivo acquisitions of fiber orientation data in the beating heart. Current methods are however limited in resolution to a few short-axis slices. For this particular application and others where the diffusion volume is subsampled, partial or even damaged, the reconstruction of a complete volume can be challenging. To address this problem, we present two complementary methods for fiber reconstruction from sparse orientation measurements, both of which derive from second-order properties related to fiber curvature as described by Maurer-Cartan connection forms. The first is an extrinsic partial volume reconstruction method based on principal component analysis of the connection forms and is best put to use when dealing with highly damaged or sparse data. The second is an intrinsic method based on curvilinear interpolation of the connection forms on ellipsoidal shells and is advantageous when more slice data becomes available. Using a database of 8 cardiac rat diffusion tensor images we demonstrate that both methods are able to reconstruct complete volumes to good accuracy and lead to low reconstruction errors.