Research On Theory And Methods For Geometric Modeling Based On Basis Functions Possessing Shape Parameters

In the last years, constructing basis functions with shape parameters is a hot research topic in Computer Aided Geometric Design (CADG), which has important theoretical significance and application value. In this doctoral dissertation, we construct quasi cubic Bernstein basis functions with two exponential shape parameters and quasi cubic trigonometric Bernstein basis functions with two exponential shape parameters in two new quasi cubic algebraic function space and quasi cubic trigonometric function space, respectively. Based on the two new basis functions, two classes of quasi cubic non-uniform B-spline basis possessing two local exponential shape parameters and quasi cubic trigonometric non-uniform B-spline basis possessing two exponential shape parameters are constructed. The proposed bases have the important properties of partition of unity, non-negativity, linear independence and total positivity and so on. The provided exponential shape parameters serve as tension shape parameters and thus they have a predictable adjusting role on the shape of the corresponding curves and surfaces. The conventional spline basis functions can be only used to generate either interpolation curves or approximation curves, in order to overcome these drawbacks, we construct a kind of quasi quartic trigonometric B-spline basis which can be used to generate interpolation curves as well as approximation curves. Shape preserving interpolation spline has great potential applications in industrial design and scientific data visualization, which has attracted widespread interest during the past thirty years. In the existing shape preserving C2interpolation spline methods, however, some methods can be only used to preserve the monotonic data set, while others can be only used to preserve the convex data set, and often for C2continuity, it is requested to solve a linear system of consistency equations for the derivative values at the knots. In this doctoral dissertation, we construct a new rational quartic interpolation spline which can achieve C continuity automatically. The main research work and achievements of the doctoral dissertation are as follows:(1) By using the blossom approach, we construct a class of quasi cubic Bernstein basis functions with two exponential shape parameters in the new quasi cubic algebraic function space Span{1,3t2-2t3,(1-t)?,l?}. Based on the new proposed quasi cubic Bernstein basis, we construct a class of quasi cubic non-uniform B-spline basis with two local exponential shape parameters. Moreover, we also extend the quasi cubic Bernstein basis functions to the triangular domain, and thus we give a new class of quasi cubic Bernstein-Bezier basis functions with three exponential shape parameters over triangular domain.The quasi cubic Bernstein basis functions include the classical cubic Bernstein basis functions and cubic Said-Ball basis funxtions as special cases. Within the general framework of Quasi Extended Chebyshev space, we prove that the quasi cubic Bernstein basis forms a normalized optimal totally positive basis. In order to compute the corresponding quasi cubic Bezier curves stably and efficiently, a new corner cutting algorithm is developed. Based on the theory of envelop and topological mapping, shape analysis for the quasi cubic Bezier curves is given. Necessary and sufficient conditions are derived for the quasi cubic Bezier curves with one or two inflection points, a loop or a cusp, and be locally or globally convex, which are completely determined by the vertexes of the control polygon and the shape parameters. We prove that the quasi cubic non-uniform B-spline basis has the important properties of partition of unity, local supportive, linear independence and total positivity and so on. The associated quasi cubic B-spline curves have C2continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be C2?FCk+3(k?Z+) continuous for particular choice of shape parameters. Based on quasi cubic Bernstein-Bezier basis functions, a class of quasi cubic Bernstein-Bezier patches are construted. A De Casteljau-type algorithm for computing the associated quasi cubic Bernstein-Bezier patches is developed. And the conditions for G1continuous joining two quasi cubic Bernstein-Bezier patches over triangular domain are also deduced.(2) A class of quasi cubic trigonometric Bernstein basis functions with two exponential shape parameters are constructed by using the blossom approach in the new functions space Span{1,sin2t,(1-sint)?,(1-cost)?}. Based on the new proposed quasi cubic trigonometric Bernstein basis, a class of quasi cubic trigonometric non-uniform B-spline basis with two local exponential shape parameters is constructed. By using the method of tensor product, a kind of rectangular trigonometric Bezier basis with four exponential shape parameters is constructed. Moreover, a class of trigonometric Bernstein-Bezier basis functions over triangular domain with three exponential shape parameters is also constructed.Within the general framework of Quasi Extended Chebyshev space, we prove that the quasi cubic trigonometric Bernstein basis forms a normalized optimal totally positive basis. In order to compute the corresponding quasi cubic trigonometric Bezier curves stably and efficiently, a new corner cutting algorithm is developed. The control points selection schemes for exactly representing any arc of an ellipse or parabola by using the quasi cubic trigonometric Bernstein basis are developed. Partition of unity, local supportive, linear independence and total positivity of the quasi cubic trigonometric non-uniform B-spline basis are proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C2?FC3continuous for non-uniform knot vector, and C3or C5continuous for uniform knot vector. The G1,G2,G3and G5continuous conditions for joining two rectangular trigonometric Bezier patches are given. Ellipsoid or paraboloid patches can be represented exactly by using the rectangular trigonometric Bezier patch. The corresponding trigonometric Bernstein-Bezier patch over triangular domain can be used to construct some surfaces whose boundaries are arcs of ellipse or parabola. The sufficient conditions for G1continuous joining two trigonometric Bernstein-Bezier patches over triangular domain are deduced.(3) Five new quasi quartic trigonometric Bernstein basis functions analogous to the classical quartic Bernstein basis function are constructed in the space Span{1, ? sint(1-sin t)?-1,?cost(1-cos t)?-1,(1-sin t)?,(1-cost)?}. Based on the quasi quartic trigonometric Bernstein basis functions, a class of quasi quartic trigonometric B-spline basis with four local shape parameters are also constructed.The ellipses and parabolas can be represented exactly by using the corresponding quasi quartic trigonometric Bezier curves. The sufficient conditions for the quasi quartic trigonometric B-spline basis having local supportive and linear independence are derived. The corresponding quasi quartic trigonometric B-spline curves have monotonic preserving property as well as convexity preserving property. And for particular choice of the shape parameters, the spline curves can be C2?FC2k+3(k?Z+) continuous. Without solving a linear system, the spline curves can be also used to approximate or interpolate sets of points with C2continuity partly or entirely.(4) A new rational quartic interpolation spline with two local tension parameters is developed.Without solving a liner system, the spline can be C2continuous. A convergence analysis establishes an error bound and shows that the order of approximation is O(h2) accuracy. By constraining the shape parameters, we derive the sufficient conditions for the proposed interpolation spline to preserve the shape of positive, monotonic, and convex set of data.