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4.3 D-NURBS Equations of Motion

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 tex2html_wrap_inline2007 be a set of generalized coordinates. These N functions of time are assembled into the vector tex2html_wrap_inline1921 . Let tex2html_wrap_inline2013 be the generalized applied force that acts on tex2html_wrap_inline2015 . We assemble the tex2html_wrap_inline2017 into the vector tex2html_wrap_inline2019 . We also assume that tex2html_wrap_inline1931 is the concatenation of N vectors tex2html_wrap_inline2025 .

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 tex2html_wrap_inline2033 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 tex2html_wrap_inline2035 mass matrix. Similarly, let tex2html_wrap_inline2037 be the damping density function. The dissipation energy is




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 tex2html_wrap_inline2039 and tex2html_wrap_inline2041 are elasticity functions which control local tension and rigidity, respectively, in the two parametric coordinate directions.gif In view of (8), the tex2html_wrap_inline2035 stiffness matrix is


where the subscripts on tex2html_wrap_inline1931 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 tex2html_wrap_inline2051 , is


and where


next up previous contents
Next: 5 Forces and Constraints Up: 4 Formulation of D-NURBS Previous: 4.2 Surfaces

Demetri Terzopoulos | Source Reference