D-NURBS: A Physics-Based Framework for Geometric Design

This paper presents dynamic NURBS or D NURBS a physics based generalization of Non Uniform Rational B Splines NURBS have become a de facto stan dard in commercial modeling systems because of their power to represent both free form shapes and common analytic shapes Traditionally however NURBS have been viewed as purely geometric primitives which require the designer to interactively adjust many degrees of free dom DOFs control points and associated weights to achieve desired shapes The conventional shape modi ca tion process can often be clumsy and laborious D NURBS are physics based models that incorporate mass distri butions internal deformation energies forces and other physical quantities into the NURBS geometric substrate Their dynamic behavior resulting from the numerical in tegration of a set of nonlinear di erential equations pro duces physically meaningful hence intuitive shape vari ation Consequently a modeler can interactively sculpt complex shapes to required speci cations not only in the traditional indirect fashion by adjusting control points and setting weights but also through direct physical ma nipulation by applying simulated forces and local and global shape constraints We use Lagrangian mechanics to formulate the equations of motion for D NURBS curves tensor product D NURBS surfaces swung D NURBS sur faces and triangular D NURBS surfaces We apply nite element analysis to reduce these equations to e cient numerical algorithms computable at interactive rates on common graphics workstations We implement a pro totype 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