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.
