Steiner and Poisson Traversal Initializations: Initial Curve Optimization for Geometric Flow-based Surface Filling

A surface-filling curve is a non-self-intersecting single curve uniformly covering a 3D object's surface. These curves are actively studied for their broad applicability. State-of-the-art generation algorithms rely on geometric flows. However, curve initialization for geometric flows has received little attention despite its significant impact on performance. Moreover, straightforward alternative initializations suffer from low robustness, slow speed, and inflexibility. To address this, we propose two complementary initialization algorithms: the ultra-fast Steiner Traversal Initialization (STI) and the highly robust Poisson Traversal Initialization (PTI). STI traverses around an approximate minimum Steiner tree on the dual graph, while PTI uses geodesic Poisson disk sampling to construct and traverse a tree structure. Extensive experiments show that while STI's performance degrades on meshes with sliver triangles, it accelerates curve generation by up to 37.14× on high-quality meshes. Meanwhile, PTI achieves up to a 32.63× speedup, operating robustly even on poorly tessellated meshes containing severe sliver triangles.