Automatic knot determination of NURBS for interactive geometric design

Abstract

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. Many geometric algorithms have been developed for NURBS during the past three decades. 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 existing techniques primarily concentrate on NURBS control points. In this paper we further augment our NURBS modeling capabilities by incorporating NURBS' non-uniform knot sequence into our shape parameter set. The automatic determination of NURBS knots will facilitate the realization of the full geometric potential of NURBS. We also have developed a modeling framework which supports a large variety of functionals ranging from simple quadratic energy forms to non-linear curvature-based (or area-based) objective functionals.