Representing a 3D shape by a set of 1D curves that are locally symmetric with respect to its boundary (i.e., curve skeletons) is of importance in several machine intelligence tasks. This paper presents a fast, automatic, and robust variational framework for computing continuous, subvoxel accurate curve skeletons from volumetric objects. A reference point inside the object is considered a point source that transmits two wave fronts of different energies. The first front (beta-front) converts the object into a graph, from which the object salient topological nodes are determined. Curve skeletons are tracked from these nodes along the cost field constructed by the second front (alpha-front) until the point source is reached. The accuracy and robustness of the proposed work are validated against competing techniques as well as a database of 3D objects. Unlike other state-of-the-art techniques, the proposed framework is highly robust because it avoids locating and classifying skeletal junction nodes, employs a new energy that does not form medial surfaces, and finally extracts curve skeletons that correspond to the most prominent parts of the shape and hence are less sensitive to noise.