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.

翻译：从样本点重建二维曲线长期以来一直是计算机图形学中的关键挑战，在矢量图形领域具有重要应用。曲面上的曲线设计与编辑直到最近才开始受到关注，主要依赖人工辅助，否则便受限于极其严格的采样条件。在本研究中，我们正式改进了现有技术的要求，提出了一种创新算法，能够根据给定的稀疏样本点集直接在曲面上重建闭合曲线。我们将最先进的平面曲线重建方法扩展并适配到曲面领域，同时解决了在非欧几里得域上工作所带来的挑战。通过在多种曲面网格上重建多条曲线，我们证明了本方法的鲁棒性。我们探索了该方法在黎曼流形上实现曲线自动化重建的新颖潜在应用。