D NURBS A Physics Based Geometric Design Framework

This paper presents dynamic NURBS or D NURBS a physics based generalization of non uniform ra tional B splines NURBS have become a de facto standard in commercial modeling systems because of their power to represent both free form and common analytic shapes Traditionally however NURBS have been viewed as purely geometric primitives which require the designer to interactively adjust many degrees of freedom DOFs control points and associated weights to achieve desired shapes The con ventional shape modi cation process can often be clumsy and laborious D NURBS are physics based models that incorporate mass distributions 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 di erential equations produces physically meaningful hence intuitive shape variation Consequently a modeler can interactively sculpt complex shapes to required speci ca tions not only in the traditional indirect fashion by adjusting control points but also through direct physical manipulation by applying simulated forces and local and global shape constraints We use La grangian mechanics to formulate the equations of motion for D NURBS curves tensor product surfaces swung surfaces and triangulated surfaces We apply nite element analysis to reduce these equations to e cient algorithms that can be simulated at interactive rates using standard numerical techniques We describe a prototype modeling environment based on D NURBS and demonstrate that D NURBS can be e ective tools in a wide range of CAGD applications such as shape blending scattered data tting and interactive sculpting