The previous two sections presented D-NURBS curve and surface geometry
in a unified way. D-NURBS physics are based on the work-energy version
of Lagrangian dynamics [14]. In an abstract physical
system, let be a set of generalized coordinates. These *N*
functions of time are assembled into the vector . Let
be the generalized applied force that acts on . We
assemble the into the vector . We also assume that is the concatenation of *N* vectors .

To proceed with the Lagrangian formulation, we will define kinetic
energy *T*, potential energy *U*, and Raleigh dissipation energy *F*
which are functions of the generalized coordinates and their
derivatives. The Lagrangian equations of motion are then expressed as

Variants of this equation have served as the basis for deformable model formulations [36]. Using (10), we can take an arbitrary geometric model, such as a NURBS, introduce appropriate kinetic, potential, and dissipation energies, and systematically formulate a physics-based, dynamic generalization of the model [17].

In the sequel, we will discuss only D-NURBS surfaces (we can consider D-NURBS curves as a special case with a simpler expression in fewer variables). To define energies and derive the D-NURBS equations of motion, let be the mass density function defined over the parametric domain of the surface. The kinetic energy of the surface is

where (using (9))

is an mass matrix. Similarly, let be the damping density function. The dissipation energy is

where

is the damping matrix.

For the elastic potential energy of D-NURBS, we can adopt the *
thin-plate under tension* energy model [35], which was
also used in [5, 42] (other energies are possible,
including the nonquadratic, curvature-based energies in
[36, 19]):

The and are elasticity functions which control local tension and rigidity, respectively, in the two parametric coordinate directions. In view of (8), the stiffness matrix is

where the subscripts on denote parametric partial derivatives.

In Appendix B, we show that by applying (10), the D-NURBS equations of motion are given by

where the generalized force vector, obtained through the principle of virtual work [14] done by the applied force distribution , is

and where