  Reconstructing 2D curves from sample points has long been a critical
challenge in computer graphics, finding essential applications in vector
graphics. The design and editing of curves on surfaces has only recently begun
to receive attention, primarily relying on human assistance, and where not,
limited by very strict sampling conditions. In this work, we formally improve
on the state-of-the-art requirements and introduce an innovative algorithm
capable of reconstructing closed curves directly on surfaces from a given
sparse set of sample points. We extend and adapt a state-of-the-art planar
curve reconstruction method to the realm of surfaces while dealing with the
challenges arising from working on non-Euclidean domains. We demonstrate the
robustness of our method by reconstructing multiple curves on various surface
meshes. We explore novel potential applications of our approach, allowing for
automated reconstruction of curves on Riemannian manifolds.
