Subdivision-based Algorithms for Surface Modeling

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Subdivision was initially conceived as an extension of splines to control meshes ofarbitrary topology. Subdivision algorithms, while being more flexible and general,retain a number of useful properties of splines. Most importantly, the surfaces generatedby these algorithms have a natural hierarchical structure and are computedusing simple local rules. This allows the design of efficient hierarchical, local, andadaptive algorithms for rendering, manipulation, collision detection and intersectionof such surfaces. Subdivision surfaces are rapidly gaining wide acceptance inthe industry. In recent years, substantial progress has been made in understandingthe properties of the subdivision surfaces, especially smoothness properties. However,subdivision still lacks the body of well understood and reliable algorithmsavailable for splines and a few practically important aspects of subdivision, suchas subdivision rules for boundaries and singular features, do not have a rigoroustheoretical foundation. Aside from smoothness, no other property of subdivisionsurfaces has been studied in detail. At the same time, some fundamental limitationsof the commonly used stationary subdivision schemes became apparent. For example,practical curvature-continuous subdivision surfaces necessarily have a "flat."spot near extraordinary vertices, and the surfaces acquire ripples near vertices ofhigh valence. The proposed research has three main objectives:Advance the theory of stationary subdivision to include all important practicalcases, and develop the theory necessary to understand fairness, meshquality, and approximation properties of subdivision.Develop robust algorithms for common geometric operations such as com-puttingcollisions and intersections of subdivision surfaces;Explore nonstationary subdivision schemes which retain locality, and identifyways to improve the quality of the resulting surfaces without sacrificingthe performance.The three components are unified by a common goal: the development of asurface representation supporting highly efficient, local, hierarchical and adaptivealgorithms for geometric modeling.