A Novel Optimization Approach to the Effective Computation of NURBS Knots

This paper presents a novel modeling technique and develops an interactive algorithm that facilitates the automatic determination of non-uniform knot vectors as well as other control variables for NURBS curves and surfaces through the unified methodology of energy minimization, variational principle, and numerical techniques. NURBS have become a de facto industry-standard primarily because of their power to represent free-form shapes as well as commonly-used analytic shapes. Although many geometric algorithms have been developed for NURBS, existing techniques primarily concentrate on NURBS control points. Recently, the optimization principle has been widely studied, which affords designers to interactively manipulate NURBS via energy functionals, simulated forces, qualitative and quantitative constraints, etc. The key advantage of energy-based approaches is that they can evolve both the control points and the weights in response to NURBS deformation resulted from the numerical optimization of a set of energy functionals. These energy functionals have the capability to quantify user-centered aesthetic criteria, qualitative constraints, and functional requirements for a large variety of applications in a unified fashion. In this paper, we further augment our NURBS modeling capabilities by incorporating NURBS’ non-uniform knot sequence into our shape parameter set intuitively controlled by energy functionals. We also have implemented a software environment that supports a large variety of functionals ranging from simple quadratic energy forms to non-linear curvature-based objective functionals.