Two Types Of Non-linear Interpolating Curve Subdivision Schemes

Curve subdivision schemes lay the foundation for understanding surface schemes and they are widely applied in computer graphics,geometric design,industrial manufacturing and other fields.In this thesis two non-linear interpolating Curve Subdivision Schemes are presented :On one hand,I discuss the incenter subdivision scheme is an interpolatory,nonlinear,shape and circle preserving curve subdivision scheme that generates G~1 curves.The limit curves of incenter subdivision scheme interpolate given points,but in general do not interpolate tangent vectors or curvatures at the given points.I change how we compute the points and associated tangent vectors that are inserted in the first step of this scheme.This modified step can be seen as a preprocessing step of the algorithm.With this modification,the algorithm generates curves in the limit that interpolate Hermite data at the given points.The theoretical analysis and numerical examples show the validity of this new incenter scheme.On the other hand,I consider a geometric two-parameter subdivision method.Such multiparameter schemes are popular because of their greater modeling flexibility which helps bringing the limiting curves interactively into a desired shape.In our scheme,the new control points are determined by the original control points and their tangents: the quadratic rational B(?)zier representation and its subdivision at the midpoint is used,where any two consecutive points and the intersection point of their tangents are the control points of the B(?)zier representation.The middle weight is taken as the first parameter ?.Then we calculate new tangent vectors at all points: after defining provisional tangent vectors by subdividing the rational quadratics,the tangent of the circle passing through this point and its two neighbors is computed;whereafter the new tangents are determined from the provisional and the circle tangents by an averaging with a second parameter?.The theoretical analysis shows that the scheme is convergent and convexity preserving.Some examples show the flexibility of the scheme and the smoothness of the curves generated by it.For special choices of the parameters,two conclusions are obtained for the geometric subdivision method.Let the second parameter ?=0,and in every step we define a new factor by the initial parameter ? such that the limiting curve is a piecewise rational quadratic C~1 curve.The circle preserving of this scheme can be obtained by computing new points with the well-known circle wights for the parameters ? in every step where ?=1.