A Primer on Shapes: Curves and Surfaces

Dynamic NURBS, or D-NURBS, are a physics-based generalization of non-uniform rational 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 conventional shape modification 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 differential equations, produces physically meaningful, hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications 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 Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product surfaces, swung surfaces, and triangulated surfaces. We apply finite element analysis to reduce these equations to efficient 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 effective tools in a wide range of CAGD applications such as shape blending, scattered data fitting, and interactive sculpting.