基函数中带形状参数的几何造型理论与方法研究
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摘要
构造带形状参数的基函数是近年来计算机辅助几何设计中的一个热门研究课题,有重要的理论意义和广阔的应用前景。本文分别在新的拟三次代数函数空间和拟三次三角函数空间中运用开花方法构造带两个指数形状参数的拟三次Bernstein基和拟三次三角Bernstein基。在此基础上,分别构造两组带两个局部指数形状参数的拟三次非均匀B样条基和拟三次三角非均匀B样条基。所构造的基函数具有单位性,非负性,线性无关性和全正性等重要性质。引入的指数形状参数具有张力作用效果,对所生成的曲线曲面形状具有明确的几何调控意义。传统的样条基只能生成逼近曲线或插值曲线,本文构造了一组既可生成逼近曲线也可生成局部插值或整体插值曲线的拟四次三角非均匀B样条基。保形插值样条在工业设计和科学数据可视化中具有重要的研究价值,在过去三十年中一直受到学者的广泛关注。但在已有的c2连续保形插值样条中,有些方法只能保单调性,有些方法只能保凸性,而且为了获得c2连续的样条,大多数方法需要求解样条在节点上满足二阶连续性的线性方程组,本文构造了一类可自动达到c2连续的四次有理保形插值样条基。本文的主要研究工作及成果如下：
(1)在拟三次代数函数空间Span{1,3t2-2t3,(1-t)α,tβ}中运用开花方法构造了一组带两个指数形状参数的拟三次Bernstein基。基于新提出的拟三次Bernstein基,构造了一类带两个局部指数形状参数的拟三次非均匀B样条基。此外,将拟三次Bernstein基推广至三角域上,构造了一类三角域上带三个指数形状参数的拟三次Bernstein-Bezier基。
拟三次Bernstein基包含经典的三次Bernstein基和三次Said-Ball基为特例。在拟扩展切比雪夫空间理论框架下,证明了该拟三次Bernstein基构成一组最优规范全正基。为了高效和稳定地计算相应的拟三次Bezier曲线,开发了一种新的割角算法。基于包络理论与拓扑映射的方法对拟三次Bezier曲线进行了形状分析,给出了曲线上含有奇点,拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形和形状参数决定。证明了拟三次非均匀B样条基具有单位性,局部支撑性,线性无关性和全正性等性质。相应的拟三次非均匀B样条曲线对单节点具有c2连续性,包含经典的三次非均匀B样条曲线为特例,且对特别的形状参数取值,曲线可以达到C2∩FCk+3(k∈Z+)阶连续性。基于拟三次Bernstein-Bezier基,给出了一类三角域上的拟三次Bernstein-Bezier曲面片。开发了一种计算三角域上拟三次Bernstein-Bezier曲面片的De Casteljau-type算法,并给出了G1光滑拼接两张三角域上拟三次Bernstein-Bezier曲面片的充分条件。
(2)在拟三次三角函数空间Span{1,sin2t,(1-sint)α,(1-cos,)β}中运用开花方法构造了一组带两个指数形状参数的拟三次三角Bernstein基。基于拟三次三角Bernstein基,构造了一类带两个局部指数形状参数的拟三次三角非均匀B样条基。利用张量积技巧,构造了一类矩形域上带四个指数形状参数的双拟三次三角Bezier基。此外,将拟三次三角Bernstein基推广至三角域上,构造了一类三角域上带三个指数形状参数的拟三次三角Bernstein-Bezier基。
在拟扩展切比雪夫空间理论框架下,证明了该拟三次三角Bernstein基构成一组最优规范全正基。开发了一种高效和稳定计算拟三次三角Bezier曲线的割角算法。给出了拟三次三角Bezier曲线精确表示任意一段椭圆弧和抛物弧的控制点选择方案。证明了新构造的拟三次三角B样条基具有单位性,局部支撑性,线性无关性和全正性等性质。相应的拟三次三角非均匀B样条曲线对单节点具有C2∩FC3连续性,且对均匀节点曲线可以达到C3甚至C5阶连续性。给出了G1,G2,G3和G5光滑拼接两张双拟三次三角Bezier曲面片的充分条件。给出了双拟三次三角Bezier曲面片精确表示椭球面片和抛物面片的控制点选择方案。基于拟三次三角Bernstein-Bezier基,构造了一类三角域上的拟三次三角Bernstein-Bezier曲面片。该曲面片能够用于生成边界曲线为椭圆弧或抛物弧的三角曲面片。开发了一种计算拟三次三角Bernstein-Bezier曲面片的De Casteljau-type算法。此外,推导出了G1光滑拼接两张三角域上拟三次三角Bernstein-Bezier曲面片的充分条件。
(3)在一类带有两个指数形状参数的拟四次三角函数空间Span{1,α sint(1-sint)α-1,βcost(1-cost)β-1,(1-sint).,(1-cost)β}中构造了一组与四次Bernstein基性质类似的拟四次三角Bernstein基。基于该拟四次三角Bernstein基,构造了一类带四个局部形状参数的拟四次三角非均匀B样条基。
由拟四次三角Bernstein基定义的拟四次三角Bezier曲线能够精确表示椭圆弧和抛物弧。给出了拟四次三角非均匀B样条基具有局部支撑性和线性无关性的充分条件。相应的样条曲线具有保单调性和保凸性,且对特别的形状参数取值,曲线可以达到C2∩FC2k+3(k∈Z+)阶连续性。利用拟四次三角非均匀B样条基,无需求解线性方程组,通过改变局部形状参数取值可灵活方便地生成逼近或插值控制点的C2连续样条曲线。
(4)构造了一类带两个局部形状参数的四次有理插值样条基。
无需求解线性方程组,该插值样条可以达到C2连续。分析了该插值样条的收敛性并给出了插值误差公式,结果表明该插值样条具有O(h2)逼近阶。通过限制两个局部形状参数取值,给出了该插值样条保正,保单调和保凸的充分条件。
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)α,lβ}. 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)α,(1-cost)β}. 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, α sint(1-sin t)α-1,βcost(1-cos t)β-1,(1-sin t)α,(1-cost)β}. 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.
(1)在拟三次代数函数空间Span{1,3t2-2t3,(1-t)α,tβ}中运用开花方法构造了一组带两个指数形状参数的拟三次Bernstein基。基于新提出的拟三次Bernstein基,构造了一类带两个局部指数形状参数的拟三次非均匀B样条基。此外,将拟三次Bernstein基推广至三角域上,构造了一类三角域上带三个指数形状参数的拟三次Bernstein-Bezier基。
拟三次Bernstein基包含经典的三次Bernstein基和三次Said-Ball基为特例。在拟扩展切比雪夫空间理论框架下,证明了该拟三次Bernstein基构成一组最优规范全正基。为了高效和稳定地计算相应的拟三次Bezier曲线,开发了一种新的割角算法。基于包络理论与拓扑映射的方法对拟三次Bezier曲线进行了形状分析,给出了曲线上含有奇点,拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形和形状参数决定。证明了拟三次非均匀B样条基具有单位性,局部支撑性,线性无关性和全正性等性质。相应的拟三次非均匀B样条曲线对单节点具有c2连续性,包含经典的三次非均匀B样条曲线为特例,且对特别的形状参数取值,曲线可以达到C2∩FCk+3(k∈Z+)阶连续性。基于拟三次Bernstein-Bezier基,给出了一类三角域上的拟三次Bernstein-Bezier曲面片。开发了一种计算三角域上拟三次Bernstein-Bezier曲面片的De Casteljau-type算法,并给出了G1光滑拼接两张三角域上拟三次Bernstein-Bezier曲面片的充分条件。
(2)在拟三次三角函数空间Span{1,sin2t,(1-sint)α,(1-cos,)β}中运用开花方法构造了一组带两个指数形状参数的拟三次三角Bernstein基。基于拟三次三角Bernstein基,构造了一类带两个局部指数形状参数的拟三次三角非均匀B样条基。利用张量积技巧,构造了一类矩形域上带四个指数形状参数的双拟三次三角Bezier基。此外,将拟三次三角Bernstein基推广至三角域上,构造了一类三角域上带三个指数形状参数的拟三次三角Bernstein-Bezier基。
在拟扩展切比雪夫空间理论框架下,证明了该拟三次三角Bernstein基构成一组最优规范全正基。开发了一种高效和稳定计算拟三次三角Bezier曲线的割角算法。给出了拟三次三角Bezier曲线精确表示任意一段椭圆弧和抛物弧的控制点选择方案。证明了新构造的拟三次三角B样条基具有单位性,局部支撑性,线性无关性和全正性等性质。相应的拟三次三角非均匀B样条曲线对单节点具有C2∩FC3连续性,且对均匀节点曲线可以达到C3甚至C5阶连续性。给出了G1,G2,G3和G5光滑拼接两张双拟三次三角Bezier曲面片的充分条件。给出了双拟三次三角Bezier曲面片精确表示椭球面片和抛物面片的控制点选择方案。基于拟三次三角Bernstein-Bezier基,构造了一类三角域上的拟三次三角Bernstein-Bezier曲面片。该曲面片能够用于生成边界曲线为椭圆弧或抛物弧的三角曲面片。开发了一种计算拟三次三角Bernstein-Bezier曲面片的De Casteljau-type算法。此外,推导出了G1光滑拼接两张三角域上拟三次三角Bernstein-Bezier曲面片的充分条件。
(3)在一类带有两个指数形状参数的拟四次三角函数空间Span{1,α sint(1-sint)α-1,βcost(1-cost)β-1,(1-sint).,(1-cost)β}中构造了一组与四次Bernstein基性质类似的拟四次三角Bernstein基。基于该拟四次三角Bernstein基,构造了一类带四个局部形状参数的拟四次三角非均匀B样条基。
由拟四次三角Bernstein基定义的拟四次三角Bezier曲线能够精确表示椭圆弧和抛物弧。给出了拟四次三角非均匀B样条基具有局部支撑性和线性无关性的充分条件。相应的样条曲线具有保单调性和保凸性,且对特别的形状参数取值,曲线可以达到C2∩FC2k+3(k∈Z+)阶连续性。利用拟四次三角非均匀B样条基,无需求解线性方程组,通过改变局部形状参数取值可灵活方便地生成逼近或插值控制点的C2连续样条曲线。
(4)构造了一类带两个局部形状参数的四次有理插值样条基。
无需求解线性方程组,该插值样条可以达到C2连续。分析了该插值样条的收敛性并给出了插值误差公式,结果表明该插值样条具有O(h2)逼近阶。通过限制两个局部形状参数取值,给出了该插值样条保正,保单调和保凸的充分条件。
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)α,lβ}. 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)α,(1-cost)β}. 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, α sint(1-sin t)α-1,βcost(1-cos t)β-1,(1-sin t)α,(1-cost)β}. 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.
引文
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