Semi-sharp edges and adaptivity for hexagonal subdivision surface schemes

Abstract

In recent research, hexagonal subdivision schemes for meshes with an arbitrary topology have been introduced. They can either be viewed on their own or be combined with their dual triangular counterparts to generate surfaces with high degrees of continuity, similar to the earlier approaches with quadrilateral schemes. For practical applications, however, a major obstacle is the lack of good algorithms to generate borders and to adaptively subdivide meshes up to user-controlled criteria. As hexagonal schemes operate on dual meshes with constantly changing positions of vertices, defining borders and corners is less straightforward than for primal schemes. In this paper, we describe an elegant method based on half polygons to create such borders, which also can be adapted to create sharp and even semi-sharp edges. Adaptive techniques are a necessity to manage the exponential growth of the number of new polygons created by the subdivision process. Existing techniques for dual subdivision schemes lead to quite difficult implementations, and would be impractical to extend to hexagonal schemes. Where earlier approaches are face centered, we view the problem in a dual way, focusing on whether or not to subdivide around the vertices of the precedent subdivision steps. This allows for crack-free tessellations with deeply subdivided portions closely bordering much coarser regions.