A new solid subdivision scheme based on box splines

Abstract

During the past twenty years, much research has been undertaken to study surface representations based on B-splines and box splines. In contrast, volumetric splines have received much less attention as an effective and powerful solid modeling tool. In this paper, we propose a novel solid subdivision scheme based on tri-variate box splines over tetrahedral tessellations in 3D. A new data structure is devised to facilitate the straightforward implementation of our simple, yet powerful solid subdivision scheme. The subdivision hierarchy can be easily constructed by calculating new vertex, edge, and cell points at each level as affine combinations of neighboring control points at the previous level. The masks for our new solid subdivision approach are uniquely obtained from tri-variate box splines, thereby ensuring high-order continuity. Because of rapid convergence rate, we acquire a high fidelity model after only a few levels of subdivision. Through the use of special rules over boundary cells, the B-rep of our subdivision solid reduces to a subdivision surface. To further demonstrate the modeling potential of our subdivision solid, we conduct several solid modeling experiments including free-form deformation. We hope to demonstrate that our box-spline subdivision solid (based on tetrahedral geometry) advances the current state-of-the-art in solid modeling in the following aspects: (1) unifying CSG, B-rep, and cell decomposition within a popular subdivision framework; (2) overcoming the shortfalls of tensor-product spline models; (3) generalizing both subdivision surfaces and free-form spline surfaces to a solid representation of arbitrary topology; and (4) taking advantage of triangle-driven, accelerated graphics hardware.