Lupa(?) Q-Bézier Curve And Triangular Patch Based On Discrete Convolution

Classical Bézier curves and surfaces provided a solid theoretical foundation for the development of free-form curves and surfaces,which became one of the most important research fields in Computer Aided Geometric Design.With the flourishing of q-calculus and(p,q)-calculus,generalized Bézier curves and surfaces involving q-integer and(p,q)integer have been developed and studied extensively.In this dissertation,we investigate the linear-time evaluation algorithms of Lupa(?) q-Bézier curves and Lupa(?)(p,q)-Bézier curves from the perspective of discrete convolution,and construct Lupa(?) q-Bézier triangular patches and Lupa(?)(p,q)-Bézier triangular patches over the triangular domain.For Lupa(?) q-Bézier curves,the discrete convolution structure of basis function sequence is obtained by the downward recursion of the pyramidal form of Lupa(?) q-Bernstein basis functions,and then the discrete convolution evaluation algorithm with linear complexity is constructed.Numerical experiments show that the discrete convolution linear evaluation algorithm runs more efficiently than the existing linear-time geometric algorithm(LTG algorithm).Different from the classical Bézier curve which has a de Casteljau algorithm,n!de Casteljau algorithms for each nth Lupa(?) q-Bézier curve can be generated by the pyramidal form of basis functions due to the commutativity of discrete convolution operation.And de Casteljau algorithms mainly use the linear combination of control points to obtain any point on the curve layer by layer recursively.In combination with the differentiation rule of discrete convolution,the discrete convolution representation of Lupa(?) q-Bernstein basis functions after derivation is deduced.The hodograph algorithm for Lupa(?) q-Bézier curves is given,which simplifies the calculation of the derivative of rational parameter curves.For Lupa(?)(p,q)-Bézier curves,the convolution generation structure of basis functions is proposed,then the discrete convolution linear evaluation algorithm and the hodograph algorithm are given.We construct the de Casteljau algorithm with explicit matrix representation,the intermediate nodes of this algorithm are Lupa(?)(p,q)-Bézier curves of low degree,and there is an explicit matrix representation between the nodes of each level and the original control point vector.Numerical experiments show that the de Casteljau algorithm with explicit matrix and the discrete convolution linear evaluation algorithm both are better than the existing linear-time geometric algorithm(LTG algorithm).The de Casteljau algorithm with explicit matrix representation is more efficient when calculating the value of curves of low degree.And the discrete convolution linear evaluation algorithm is more efficient when calculating the value of curves of high degree.Further generalizing two univariate basis functions to triangular domain,the bivariate Lupa(?) q-Bernstein basis functions and Lupa(?)(p,q)-Bernstein basis functions are constructed by using discrete convolution operation of double-indexed function sequences,thus generating the corresponding Lupa(?) q-Bézier triangular patch and Lupa(?)(p,q)-Bézier triangular patch.The two triangular patches preserve the excellent geometric properties of the classical Bézier triangular patches,such as convex hull,affine invariance and corner points properties,and have degree evaluation and de Casteljau algorithm.In particular,one of the boundaries of both triangular patches is a classical Bézier curve,and the other two boundaries are the corresponding Lupa(?) q-Bézier curves or Lupa(?)(p,q)-Bézier curves.In the conic sections,the boundary curve of quadratic Lupa(?) q-Bézier triangular patch and Lupa(?)(p,q)-Bézier triangular patch can be represented as a parabola and two hyperbolas,and when degenerated to a quadratic classical Bézier triangular patch,it can be expressed as three parabolas.In terms of shape control,changing the control net or changing the values of p,q can achieve the shape control of triangular patch.