Stationary and Non-Stationary Subdivision Schemes and Their Applications

Abstract

Subdivision is an easy and well-defined method to describe smooth curves and surfaces. Its application ranges from industrial design and animation to scientific visualization and simulation. This dissertation presents a variety of stationary and non-stationary interpolating and approximating subdivision schemes with shape parameters. The proposed families generalize the several schemes, previously proposed in the literature, are shown to be members of the family. Order of continuity, curvature, error bounds, deviation error and basic limit functions for several members of the family are computed. Moreover, these schemes are shown to outperform in several aspects comparative to the similar schemes previously proposed to the literature. The non-stationary schemes are based on sinusoidal functions and continuity properties are prove by asymptotic equivalence with stationary counter parts. A comparison between the proposed non-stationary schemes and their stationary counter parts shows the former to have better curvature behavior. It is proved that the limiting conic sections generated by proposed non-stationary schemes have less deviation from being the exact conic sections. Moreover, proposed 3-point ternary schemes with fewer initial control points produced better limiting conic sections than other existing schemes. Further the fractal behavior of binary interpolating subdivision schemes has been discussed. The association between the fractal behavior of the limit curve and the surface with the tension parameter is also elaborated. Some families of the schemes are constructed by fitting multivariate vi polynomial functions of any degree to different types of data by least square techniques. Furthermore, it is straightforward to construct schemes for fitting data in higher dimensional spaces by using proposed framework.