Research On The Refinability Of Cubic Parametric Splines With Tension Properties

Splines,as well as refinable functions,have been widely used in the numerical solution of differential equations,computer-aided geometric design(CAGD) and wavelet analysis and so on.Since 1980s-1990s,with the theory of wavelet analysis being increasingly ripe and widely used,multi-scale idea and multi-scale equations,which in the process of configuration of wavelet,make people pay attention to another natural property of B-spline-refining,resulting in curves and surfaces of subdivision expression.Refinable functions form the foundation for the theory of compactly supported wavelets and the theory of subdivision schemes.However,curves and surfaces by refinable parametric splines subdivision are important members of subdivision curves and surfaces,which help solve new problems that NURBS faces in the practical application,such as the problems of the choice of weights and parameters.In recent years,scholars,who are at home and abroad,constructed kinds of expansion of B-spline models with shape parameters.On the one hand,parameters were introduced in the allocation functions;on the other hand,shape parameters were introduced in splines based on piecewise cubic components,by changing the value of the parameters to adjust the shape of curves.However,these models generally lacked of the essence of spline's refinability,and couldn't express curves and surfaces' discrete generation.In 2004,C.Marnni constructed cubic parametric splines with tension properties,which contained classical C~2 cubic B-splines.He further researched on refining of splines,and presented the binary refinement equation in case of uniform nodes,which provided a new idea and method that in constructing spline functions with shape parameters and studying on refining.This paper bases on the previous studies,discusses thoroughly refining of cubic parametric splines with tension properties,focuses on ternary refinement equations,further extendes to the M-band refinement equation,where(2≤M≤5,M∈N).It provides new ideas and methods in applying parametric splines with tension properties and further studying on refining.The main research results are in this article as follows:First,we focus on binary refinement matrix of cubic parametric splines with tension properties in case of non-uniform nodes,including the calculation of tension parameters and refinement matrix when selecting a set of different nodes.We analyz the results,and further give binary subdivision rules.The results of this article contains the results of C.Manni (binary refinement under the uniform node sequences).Second,we study ternary refinement process of cubic parameters splines with tension parameters,give explicit expressions of tension parameters in refinement layer and refinement matrix,and summariz ternary subdivision rules to facilitate the direct application. Comparing with binary subdivision,ternary subdivision curves have advantages in approach speed.Third,we further extend the research results to M-band refinement,give the explicit expression of five- band refinement matrix,where(2≤M≤5,M∈N).