Two Classes Of Generalized Ball Curves And Surfaces Based On Q-Calculus

Bernstein polynomial and B(?)zier method are the foundations of parametric curve and surface modeling in Computer Aided Geometric Design.Ball curve and surface and its extension form provide more choices for more efficient calculation of points on parametric curve and surface.Phillips q-B(?)zier curve is a generalized model of B(?)zier curve in the sense of q-calculus.Based on q-calculus theory,combining Said-Ball basis function,Wang-Ball basis function and Phillips q-Bernstein basis function,the generalized models of Said-Ball curve and surface and Wang-Ball curve and surface for any order are obtained in the sense of q-calculus.The main research achievements of this dissertation are as follows:Firstly,based on the q-calculus theory and combining the Said-Ball and q-Bernstein polynomials,the Said-Ball basis function containing q-integer is constructed,namely,q-Said-Ball basis function.The n-degree q-Said-Ball basis function has the same excellent properties as Said-Ball basis function: non-negativity,partition of unity,end-point property,etc.When q equals 1,it reduces to Said-Ball basis function of degree n.By constructing the corner cutting algorithm,it is proved that the odd degree q-Said-Ball basis function has totally positivity,and the conversion formula between q-Said-Ball basis functions is given.q-Said-Ball curve is constructed based on q-Said-Ball basis function,and its basic properties and recursive evaluation algorithm are studied.q-Said-Ball curve of degree(28)+ 1)can be transformed into Phillips q-B(?)zier form of(8)+ 1)after m times recursive iteration.q-Said-Ball curve of degree 2m can be degree elevated once and repeat the same operation for the odd-degree curve.Secondly,inspired by q-calculus and q-Bernstein polynomial,a generalized model of arbitrary degree Wang-Ball curve with shape parameter q is proposed,namely,q-Wang-Ball curve.We defined q-Wang-Ball basis function,studied and proved its basic properties,such as non-negative,partition of unity,end-point property and recurrence relation of basis function,etc.Unlike q-Said-Ball basis function,q-Wang-Ball basis function is not totally positive basis.Based on q-Wang-Ball basis function,the definition and recursive evaluation algorithm of q-Wang-Ball curve are given.It is found that the calculation of n-order q-Wang-Ball curve can be converted into the calculation of cubic or quadratic q-Wang-Ball curve.The algorithm efficiency and time complexity of these two generalized q-Ball curves are calculated and compared with q-B(?)zier curve.The results show that q-Said-Ball is better than q-B(?)zier and q-Wang-Ball is better than q-Said-Ball in the evaluation and the calculation speed of degree elevation and reduction.In addition,q-Said-Ball curve and q-Wang-Ball curve are extended to the case of tensor product,and q-Said-Ball surface and q-Wang-Ball surface of tensor product type are obtained.The generalized q-Ball surfaces are used to replace q-B(?)zier surface in geometric design and graphic rendering,so as to reduce frequent and large number of calculations.