Surface Covering Curves

Abstract

In this work, we present the concept of surface covering curves that can be used to construct wire sculptures or surface textures. We show that any mesh surface can be converted to a single closed 3D curve that follows the shape of the mesh surface. We have developed two methods to construct corresponding 3D ribbons and yarns from the mesh structure and the connectivity of the curve. The first method constructs equal thickness ribbons (or equal diameter yarns). The second method creates ribbons with changing thickness (or yarns with changing diameter) that can densely cover the mesh surface. Since each iteration of any subdivision scheme results in a denser mesh, the procedure outlined above can be used to obtain a denser and denser curve. These curves can densely cover a mesh surface in limit. Therefore, this approach along with a subdivision scheme provides visual results that are similar to space filling curves that are created by fractal algorithms. Unlike space filling curves which fills a square or a cube, our curves cover a surface, and henceforth, we called them "surface covering curves". Space covering curves also resemble TSP (traveling salesmen problem) art and Truchet-like curves that are embedded on surfaces.