Dynamic Modeling of Butterfly Subdivision Surfaces

ÐIn this paper, we develop integrated techniques that unify physics-based modeling with geometric subdivision methodology and present a scheme for dynamic manipulation of the smooth limit surface generated by the (modified) butterfly scheme using physics-based aforceo tools. This procedure-based surface model obtained through butterfly subdivision does not have a closed-form analytic formulation (unlike other well-known spline-based models) and, hence, poses challenging problems to incorporate mass and damping distributions, internal deformation energy, forces, and other physical quantities required to develop a physics-based model. Our primary contributions to computer graphics and geometric modeling include: 1) a new hierarchical formulation for locally parameterizing the butterfly subdivision surface over its initial control polyhedron, 2) formulation of dynamic butterfly subdivision surface as a set of novel finite elements, and 3) approximation of this new type of finite elements by a collection of existing finite elements subject to implicit geometric constraints. Our new physics-based model can be sculpted directly by applying synthesized forces and its equilibrium is characterized by the minimum of a deformation energy subject to the imposed constraints. We demonstrate that this novel dynamic framework not only provides a direct and natural means of manipulating geometric shapes, but also facilitates hierarchical shape and nonrigid motion estimation from large range and volumetric data sets using very few degrees of freedom (control vertices that define the initial polyhedron). Index TermsÐDynamic modeling, physics-based geometric design, geometric modeling, CAGD, subdivision surfaces, deformable models, finite elements, interactive techniques.