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：从样本点重构二维曲线一直是计算机图形学中的关键挑战，在矢量图形中具有重要应用。曲面上的曲线设计与编辑直至近期才开始受到关注，主要依赖人工辅助，即便无需人工干预，也受限于极为严格的采样条件。本文在现有技术要求的基础上取得正式突破，提出一种创新算法，能够根据给定的稀疏样本点集直接在曲面上重构闭合曲线。我们将先进的平面曲线重构方法进行扩展与适配，使其适用于曲面领域，同时解决非欧几里得域计算带来的挑战。通过在多种曲面网格上重构多条曲线，我们验证了该方法的鲁棒性。我们探索了该方法的新颖应用前景，实现了黎曼流形上曲线的自动化重构。