Two Classes Of Curve Modeling Methods Based On Iteration

Curve and surface modeling is one of the main research contents in Computer Aided Ge-ometric Design. Data fitting is a vital tool of curve and surface modeling, which constructs afitting curve or patch by fitting a series of data points. Concern about the relationship betweenthe original data and the final fitting curves and surfaces, the fitting methods can be distinguishedinto two types: interpolation and approximation. This paper constructs two types methods to fitdiscrete data points based on multi-stage subdivision and progressive iteration approximationquality of cubic NURBS curves.Multi-stage subdivision breaks the whole process into several simple and highly localstages, it is a iteratively repeating process and leads eventually to a limit fitting curve or patch.With the help of Rechardson’s extrapolation principle, classical four-point and six-point sten-cils, and based on the six-point multi-step subdivision, this paper proposes four kinds of inter-polating multi-step subdivision schemes, by manipulating the former two stages of the threesubdivision steps.Compared with the classical six-point scheme, the new schemes have higherHolder continuity and more smoothing curvature distribution. Relaxing the interpolation con-dition, we design an approximating scheme with an shape parameter. This scheme has bettergeometric property when the parameter is in a particular range.Progressive iteration approximation is a new data fitting technology in the recent dozensof years. Taking the given points as the initial control points and constructing a cubic B-splinecurve. PIA constructs a series of iterative limit curves which have finer and finer fitting pre-cisions by iteratively adjusting the points. If the limit fitting curve interpolation on the initialcontrol points, the curve holds PIA property. This paper proposes PIA fitting method usingcubic NURBS curve under centripetal parameterization. This parameterization take the curves’bend into account and can yield better results when the geometric distribution of control pointsare dramatically uneven. Furthermore, the cubic NURBS curve holds local PIA property, andwe design local PIA data fitting method using centripetal parameterization. On one hand, thedata points can be fitted one by one. On the other hand, the fitting precision can be controlledseparately. In other words, the data points can be fitted under varying fitting precisions.