Research On Interproximate Subdivision Curves Modeling

In the last 40 years,subdivision has been one of the research hotspots in CAGD,CAD&CG.Subdivision has also been implanted in curve/surface reconstruction due to its inherent properties.In general,subdivision schemes can be classified into two main types:interpolating and approximating subdivision schemes.Interpolating schemes define rules which pass through original points exactly and also generate new points according needs.Approximating schemes define rules which can approximate original points well.In the existing subdivision scheme,if we want to achieve the interpolation,approximation,or both of the interpolation and approximation of the control point,the combined scheme is a more appropriate choice.However,for combined scheme,the process of adjusting parameters is very complex.Thus,it is necessary to construct a new subdivision framework to avoid the complicated parameter adjustment process and realize the flexible processing for data points.This thesis mainly discusses the general framework of non-stationary interproximate subdivision scheme,and the analysis of the important properties.The main contributions are as follows.1.A family of non-stationary interproximate subdivision schemes are proposed and the convexity preservation has been further discussed.It provides a binary subdivision schemes that connects interpolating schemes with approximating schemes.Data points for art and design are divided into interpolating points and approximating points by specific rules.Thus,the limit curves of our schemes can interpolate feature points and approximate the other points simultaneously.Comparisons show that our schemes are more flexible in dealing with data points and the final effects are more faithful to designers'ideas.2.To achieve interpolation,approximation or interproximation for control point set,a new non-stationary binary subdivision framework has been established.Starting from a non-stationary binary combined subdivision scheme firstly,then data points have been divided into interpolating point set and approximating point set.And then,the non-stationary geometric rules of new edge points and new vertex points in each step are recursively defined.Finally,a new subdivision framework mixing four types of subdivision subframeworks is established.This framework considers different characteristics of interpolation,approximation and interproximation subdivision schemes synthetically.For the interproximation subdivision subframework,a set of selection rules for interpolating points have been constructed further.And the range of parameters when the subframework reachesC~2 continuity is also analyzed.3.Starting from a combined subdivision scheme,a new stationary binary interproximate subdivision framework has been established by recursively defining geometric rules of new edge points and new vertex points in each step.It is worth mentioning that the data points here have been divided into interpolating point set and approximating point set.The framework not only contains some existing classical subdivision schemes,but also includes some brand-new subdivision schemes.The subdivision schemes of the framework can interpolate the given points and approximate the other points simultaneously,and the limit curves areC~2 continuous.