D-NURBS - A PHYSICS-BASED FRAMEWORK FOR GEOMETRIC DESIGN

This paper presents dynamic NURBS, or D-NURBS, a physics-based general ization of Non-Uniform Rational B-Splines. NURBS have become a de fact o standard in commercial modeling systems because of their power to re present both free-form shapes and common analytic shapes. Traditionall y, however, NURBS have been viewed as purely geometric primitives, whi ch require the designer to interactively adjust many degrees of freedo m (DOFs)-control points and associated weights-to achieve desired shap es. The conventional shape modification process can often be clumsy an d laborious. D-NURBS are physics-based models that incorporate mass di stributions, internal deformation energies, forces, and other physical quantities into the NURBS geometric substrate. Their dynamic behavior , resulting from the numerical integration of a set of nonlinear diffe rential equations, produces physically meaningful, hence intuitive sha pe variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirec t fashion, by adjusting control points and setting weights, but also t hrough direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to for mulate the equations of motion for D-NURBS curves, tenser-product D-NU RBS surfaces, swung D-NURBS surfaces, and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficie nt numerical algorithms computable at interactive rates on common grap hics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a w ide range of CAGD applications such as shape blending, scattered data fitting, and interactive sculpting.