基于双曲函数的H-Bézier和Ferguson曲线
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摘要
本文一共包含五章内容。
第一章简单的介绍了研究背景以及主要研究内容。
第二章在空间T_n=span{1，t，t~2，…，t~(n-4)，sinht，cosht，tsinht，tcosht}中提出一组名为H-Bézier的基，讨论了该基的性质。并用该组基定义了H-Bézier曲线，同时证明有许多实际应用价值的曲线(如代数曲线和超越曲线)可以用H-Bézier曲线的形式精确表示。
第三章在第二章的基础上在空间T_n=span{1，t，t~2，…，t~(n-4)，sinht，cosht，sinh2t，cosh2t}中也提出一组名为H-Bézier的基，讨论了该基的性质。并用该组基定义了H-Bézier曲线，同时证明有许多实际应用价值的曲线(如代数曲线和超越曲线)，可以用H-Bézier曲线的形式精确表示。
第四章介绍三次样条函数的定义，给出用型值点处的一阶导数、二阶导数表示插值三次样条曲线的关系式，最后给出求解插值三次样条曲线的步骤。
第五章从双曲函数出发，构造了一类插值于首、末端点及其切矢的参数样条曲线，称为H-Ferguson曲线，并研究了合成H-Ferguson曲线的算法。引入了参数α可调整整条曲线。H-Ferguson曲线丰富了参数样条曲线，是一种可行的算法。
This thesis is composed of five chapters.
In the first chapter, the author briefly introduces the background and the maincontent of this thesis.
The second chapter, a basis called H-Bezier basis, is constructed for thespace T_n=span{1,t,t~2,…,t~(n-4,)sinh t, cosh t, t sinh t, t cosh t}. Properties of this basis areanalyzed and H-Bezier curves are defined based on it. Then it is demonstrated thatmany valuable curves, such as algebraic curves and transcendental curves, can beexpressed by H-Bezier curves.
Basis on the second chapter, in the third chapter a basis called H-Bezier basis, isconstructed for the space T_n=span{1,t,t~2,…,t~(n-4),sinh t, cosh t, sinh 2t, cosh 2t}.Properties of this basis are analyzed and H-Bezier curves are defined based on it.Then it is demonstrated that many valuable curves, such as algebraic curves andtranscendental curves, can be expressed by H-Bezier curves.
The fourth chapter, the definition of cubic spline function is introduced. By firstderivative of given data points、second derivative of given data points, therelational expression of interpolation cubic spline curve is given. Finally, the stepsof solving the interpolation cubic spline curve is given.
The fifth chapter, a kind of parameter spline curve is constructed based onalgebra and hyperbola function, which is named H-Ferguson curve. TheH-Ferguson curve interpolates start point and end point, and also interpolatestangent vectors of start point and end point. Then the arithmetic of composedH-Ferguson curve is studied. H-Ferguson curve enriches the theory of parameterspline curves, which is a feasible method of constructing interpolation curve.
第一章简单的介绍了研究背景以及主要研究内容。
第二章在空间T_n=span{1，t，t~2，…，t~(n-4)，sinht，cosht，tsinht，tcosht}中提出一组名为H-Bézier的基，讨论了该基的性质。并用该组基定义了H-Bézier曲线，同时证明有许多实际应用价值的曲线(如代数曲线和超越曲线)可以用H-Bézier曲线的形式精确表示。
第三章在第二章的基础上在空间T_n=span{1，t，t~2，…，t~(n-4)，sinht，cosht，sinh2t，cosh2t}中也提出一组名为H-Bézier的基，讨论了该基的性质。并用该组基定义了H-Bézier曲线，同时证明有许多实际应用价值的曲线(如代数曲线和超越曲线)，可以用H-Bézier曲线的形式精确表示。
第四章介绍三次样条函数的定义，给出用型值点处的一阶导数、二阶导数表示插值三次样条曲线的关系式，最后给出求解插值三次样条曲线的步骤。
第五章从双曲函数出发，构造了一类插值于首、末端点及其切矢的参数样条曲线，称为H-Ferguson曲线，并研究了合成H-Ferguson曲线的算法。引入了参数α可调整整条曲线。H-Ferguson曲线丰富了参数样条曲线，是一种可行的算法。
This thesis is composed of five chapters.
In the first chapter, the author briefly introduces the background and the maincontent of this thesis.
The second chapter, a basis called H-Bezier basis, is constructed for thespace T_n=span{1,t,t~2,…,t~(n-4,)sinh t, cosh t, t sinh t, t cosh t}. Properties of this basis areanalyzed and H-Bezier curves are defined based on it. Then it is demonstrated thatmany valuable curves, such as algebraic curves and transcendental curves, can beexpressed by H-Bezier curves.
Basis on the second chapter, in the third chapter a basis called H-Bezier basis, isconstructed for the space T_n=span{1,t,t~2,…,t~(n-4),sinh t, cosh t, sinh 2t, cosh 2t}.Properties of this basis are analyzed and H-Bezier curves are defined based on it.Then it is demonstrated that many valuable curves, such as algebraic curves andtranscendental curves, can be expressed by H-Bezier curves.
The fourth chapter, the definition of cubic spline function is introduced. By firstderivative of given data points、second derivative of given data points, therelational expression of interpolation cubic spline curve is given. Finally, the stepsof solving the interpolation cubic spline curve is given.
The fifth chapter, a kind of parameter spline curve is constructed based onalgebra and hyperbola function, which is named H-Ferguson curve. TheH-Ferguson curve interpolates start point and end point, and also interpolatestangent vectors of start point and end point. Then the arithmetic of composedH-Ferguson curve is studied. H-Ferguson curve enriches the theory of parameterspline curves, which is a feasible method of constructing interpolation curve.
引文
[1] Verprille, K.J., 1972. Computer aided design application of rational B-spline approximation from Ph.D thesis, Syracuse University, Syracuse, N.Y.
[2] Pieg, L., Tiller, W., 1989. Curve and surface constructions using rational B-splines. Computer-Aided Design [J] 19(9): 485-498.
[3] Farin, G.., 1989. Rational curves and urfaces, in: Lyche, T., Schumaker, L.(Eds), Mathematical Methods in Computer Aided Geometric Design, Academic Press, Boston.
[4] Pieg, L., Tiller, W., 1997. The NURBS Book, 2nd edn. Springer.
[5] Verprille, S.M., 1991. CAD Surface data exchange using STEP. Computer Aided Design[J], 23: 269-281.
[6] 王国瑾，汪国昭，郑建民，2001．计算机辅助几何设计．高等教育出版社，施普林格出版社．
[7] Farin, G.., 1992. From conics to NURBS; A tutorial and survey. IEE Computer Graphics and Applications[J](5): 78-86.
[8] Juhasz, I., 1995. Approximating the helix with rational cubic Bezier curves. Computer-Aided Design 27(8): 587-593.
[9] Seemann, G., 1997. Approximating a helix segment with a rational Bezier curve. Computer Aided Geometric Design14(5): 475-490.
[10] Farouki, R.T., Sakkalis, T, J., 1991. Real rational curves are not unit speed. Computer Aided Geometric Design8(2): 151-157.
[11] Carnier, J.M., Pena, J.M., 1993. Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1, 173-196.
[12] Carnier, J.M., Pena, J.M., 1994. Totally positive bases for shape preserving curve design and optimality of B-splines. Computer Aided Geometric design 11, 635-656.
[13] Pena, J.M., 1997. Shape preserving representations for trigonometric polynomial curves. Computer Aided Geometric Design 14, 5-11.
[14] Mainar, E., Pena, J.M., 1998. Evaluation and subdivision algorithms for totally positive systems, in: Daehlen, M., Lyche, T., Schumaker, T.(Eds.), Mathematical Methods for Curves and Surfaces Ⅱ, Vanderbilt University Press, Nashville, pp.327-334.
[15] Pena, J.M., 1999. Shape Preserving Representations in Computer-Aided Geometric Design. Nova Science Publishers, Commack, NY.
[16] Pena, J.M., 1999. Stability and error analysis of shape preserving representations, in: Pena, J.M., (Ed), Shape Preserving Representations in Computer-Aided Geometric Design, Nova Science, New York, pp.85-97.
[17] Schmeltz, G., 1992. Variationsreduzierende Kurvendarstellungen und Krummungskriterien fur Bezier flachen, Thesis. Fachbereich Mathematik, Technische Hochschule Darmstadt.
[18] Sanchez-Reyes, J., 1998. Harmonic rational Bezier curves, p-Bezier curves and trigonometric polynomials. Computer Aided Geometric Design 11(9): 909-924.
[19] Zhang, J., 1996. C-curves: An extension of cubic curves. Computer Aided Geometric Design 13(9): 199-217.
[20] Zhang, J., 1997. Two different forms of C-B-splines. Computer Aided Geometric Design 14(1): 31-41.
[21] Chen, Q., Wang, G..., 2003. A class of Bezier-like curves. Computer Aided Geometric Design 20(1): 29-39.
[22] 丁敏，汪国昭．基于三角和代数多项式的T-Bezier曲线[J]．计算机学报．2004，27(8)：1021-1026．
[23] 苏本跃，盛敏．基于双曲函数的Bezier型曲线曲面[J]．计算机工程与设计，2006，27(3)：370-372．
[24] 苏本跃，黄有度．T-B样条曲线及其应用[J]大学数学，2005，21(1)：87-90．
[25] 苏本跃，黄有度．CAGD中三角多项式曲线的构造及应用[J]．合肥工业大学学报：自然科学版，2005，28(1)：105-108．
[26] 苏本跃，盛敏．基于三角函数的BeZIER曲线曲面造型方法[J]．微电子与计算机，2004，21(12)：73-75，80．
[27] 王媛，康宝生．代数双曲混合H-Bezier函数及其性质[J]．西北大学学报：自然科学版．2006，36(5)：693-697．
[28] Ferguson, J.C., Multivariable Curve interpolation[J]. Journal ACM, 1964, 11(2): 199-217.
[29] 李军成，宋来忠．一类基于三角函数的插值曲线[J]．计算机与数字工程，2006，34(4)：61-63．
[30] 苏步青，刘鼎元．计算几何[M]．上海：上海科学技术出版社，1981．
[31] 朱心雄等．自由曲线曲面造型技术[M]．北京：科学出版社，2000．
[2] Pieg, L., Tiller, W., 1989. Curve and surface constructions using rational B-splines. Computer-Aided Design [J] 19(9): 485-498.
[3] Farin, G.., 1989. Rational curves and urfaces, in: Lyche, T., Schumaker, L.(Eds), Mathematical Methods in Computer Aided Geometric Design, Academic Press, Boston.
[4] Pieg, L., Tiller, W., 1997. The NURBS Book, 2nd edn. Springer.
[5] Verprille, S.M., 1991. CAD Surface data exchange using STEP. Computer Aided Design[J], 23: 269-281.
[6] 王国瑾，汪国昭，郑建民，2001．计算机辅助几何设计．高等教育出版社，施普林格出版社．
[7] Farin, G.., 1992. From conics to NURBS; A tutorial and survey. IEE Computer Graphics and Applications[J](5): 78-86.
[8] Juhasz, I., 1995. Approximating the helix with rational cubic Bezier curves. Computer-Aided Design 27(8): 587-593.
[9] Seemann, G., 1997. Approximating a helix segment with a rational Bezier curve. Computer Aided Geometric Design14(5): 475-490.
[10] Farouki, R.T., Sakkalis, T, J., 1991. Real rational curves are not unit speed. Computer Aided Geometric Design8(2): 151-157.
[11] Carnier, J.M., Pena, J.M., 1993. Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1, 173-196.
[12] Carnier, J.M., Pena, J.M., 1994. Totally positive bases for shape preserving curve design and optimality of B-splines. Computer Aided Geometric design 11, 635-656.
[13] Pena, J.M., 1997. Shape preserving representations for trigonometric polynomial curves. Computer Aided Geometric Design 14, 5-11.
[14] Mainar, E., Pena, J.M., 1998. Evaluation and subdivision algorithms for totally positive systems, in: Daehlen, M., Lyche, T., Schumaker, T.(Eds.), Mathematical Methods for Curves and Surfaces Ⅱ, Vanderbilt University Press, Nashville, pp.327-334.
[15] Pena, J.M., 1999. Shape Preserving Representations in Computer-Aided Geometric Design. Nova Science Publishers, Commack, NY.
[16] Pena, J.M., 1999. Stability and error analysis of shape preserving representations, in: Pena, J.M., (Ed), Shape Preserving Representations in Computer-Aided Geometric Design, Nova Science, New York, pp.85-97.
[17] Schmeltz, G., 1992. Variationsreduzierende Kurvendarstellungen und Krummungskriterien fur Bezier flachen, Thesis. Fachbereich Mathematik, Technische Hochschule Darmstadt.
[18] Sanchez-Reyes, J., 1998. Harmonic rational Bezier curves, p-Bezier curves and trigonometric polynomials. Computer Aided Geometric Design 11(9): 909-924.
[19] Zhang, J., 1996. C-curves: An extension of cubic curves. Computer Aided Geometric Design 13(9): 199-217.
[20] Zhang, J., 1997. Two different forms of C-B-splines. Computer Aided Geometric Design 14(1): 31-41.
[21] Chen, Q., Wang, G..., 2003. A class of Bezier-like curves. Computer Aided Geometric Design 20(1): 29-39.
[22] 丁敏，汪国昭．基于三角和代数多项式的T-Bezier曲线[J]．计算机学报．2004，27(8)：1021-1026．
[23] 苏本跃，盛敏．基于双曲函数的Bezier型曲线曲面[J]．计算机工程与设计，2006，27(3)：370-372．
[24] 苏本跃，黄有度．T-B样条曲线及其应用[J]大学数学，2005，21(1)：87-90．
[25] 苏本跃，黄有度．CAGD中三角多项式曲线的构造及应用[J]．合肥工业大学学报：自然科学版，2005，28(1)：105-108．
[26] 苏本跃，盛敏．基于三角函数的BeZIER曲线曲面造型方法[J]．微电子与计算机，2004，21(12)：73-75，80．
[27] 王媛，康宝生．代数双曲混合H-Bezier函数及其性质[J]．西北大学学报：自然科学版．2006，36(5)：693-697．
[28] Ferguson, J.C., Multivariable Curve interpolation[J]. Journal ACM, 1964, 11(2): 199-217.
[29] 李军成，宋来忠．一类基于三角函数的插值曲线[J]．计算机与数字工程，2006，34(4)：61-63．
[30] 苏步青，刘鼎元．计算几何[M]．上海：上海科学技术出版社，1981．
[31] 朱心雄等．自由曲线曲面造型技术[M]．北京：科学出版社，2000．