CAGD中三角多项式曲线曲面造型的研究
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摘要
本文主要对参数曲线曲面造型的一种新方法——三角多项式曲线曲面进行了深入研究，其内容主要包括T-Bézier曲线曲面、T-B样条曲线曲面、TC-Bézier曲线曲面和TC-B样条曲线曲面。文章最后在双曲函数空间中讨论了HC-Bézier曲线曲面。
本文首先回顾了曲线曲面造型方法的分类以及各自的特点。阐述了CAGD中参数曲线曲面造型的发展历史并介绍了Bzier方法、B样条方法以及非多项式曲线曲面造型方法，后者包括L-样条、螺旋样条、张力样条以及C-曲线等。
文章以Bézier曲线和B样条曲线的特点为基础，在三角函数空间中构造一组具有上述两类曲线特性的三角函数多项式曲线，称其为T-Bézier曲线和T-B样条曲线。它们继承了Bézier曲线和B样条曲线的特点，曲线表示简单、直观。此外由于它们还具有三角函数的优点，故既可以精确表示直线段、二次多项式曲线段又可以精确表示圆弧、椭圆弧等二次曲线以及心脏线、双纽线等超越曲线。特别地，3次均匀T-B样条曲线曲面比同阶均匀B样条(C-B样条)曲线曲面具有更高的光滑度。3次T-Bézier曲线在光滑拼接时也可以达到更高的连续性。最后由于这两类曲线仅由三角函数构成，所以它们较易转化为有理多项式曲线。从而融入到现有的几何造型系统中。
在C-曲线的启示下，本文进一步在T-Bézier曲线及T-B样条曲线中引入控制参数α用以调整曲线形状，构造了另一类自由参数曲线，称其为TC-Bézier曲线及TC-B样条曲线。这两类曲线一方面具有T-Bézier曲线及T-B样条曲线的类似性质和相关二次曲线的精确表示，另一方面由于参数α的引入使得曲线具有更强的表现能力。
文章最后运用同样的方法在双曲函数空间中构造了HC-Bézier曲线。该曲线与TC-Bézier曲线的性质完全类似，此外它既能精确表示直线段又能精确表示双曲线。HC-Bézier曲线中同样具有控制参数α，从而调整曲线形状更加灵活。
对于每一类曲线作者均将它们直接推广到张量积曲面，这些曲面可以精确表示球面、椭球面、双曲面等二次曲面。
This paper summaries the researches on the new schemes of parameter curves and surfaces modeling-curves and surfaces modeling of trigonometric polynomial, which includes curves and surfaces of T-Bezier, T-B-spline, TC-Bezier and TC-B-spline. HC-Bé zier curves and surfaces are also discussed in the space of hyperbolic functions in the end.
After briefly reviewing the classification and respective characters of curves and surfaces modeling, the paper expatiates its history in CAGD. And then we introduce Bezier, B-spline and non-polynomial curves and surfaces modeling, which include L-splines, helix splines, splines in tension and C-curves etc.
By analyzing the characters of Bezier curves and B-spline curves, we construct trigonometric polynomial curves in the space of trigonometric functions, which assume the characters of Bézier curves and B-spline curves. Their representatives are simple and direct. We call them as T-Bezier curves and T-B-spline curves. They not only inherit the advantages of Bezier curves and B-spline curves, also can be used to represent straight lines precisely and some remarkable transcendental curves precisely, such as circular arc, ellipse, cardioids and twisted pair line etc. Especially, the uniform T-B-spline curve of three degrees is smoother than B-spline curve and C-B-spline curve of the same order. When connected smoothly, T-Bezier curve of three degrees can attain more superior continuity. At last, the T-Bézier curves and T-B-spline curves can be converted to rational curves so easily that they can be merged into current geometry modeling rapidly.
Furthermore, with the illuminating of C-curves, the paper construct another freeform parameter curves and surfaces by introducing parameter a in control. We call them as TC-Bezier curves and TC-B-spline curves. On the one hand, they possess the similar characters of T-Bezier curves and T-B-spline curves. On the other hand, by introducing the parameter a, they act the wonderful ability of representation.
At last, we construct hyperbolic polynomial curves in the space of hyperbolic functions. We call them as HC-Bezier curves. Similarly, they not only can represent straight lines precisely, but also can represent quadric curve precisely, such as hyperbola and so on. Parameter a in control is also used in this kind of curve.
The generation of tensor product surfaces of every kind of curves is straightforward; these corresponding tensor product surfaces also contain many special surfaces, including spherical surface, ellipsoid, hyperboloid etc.
本文首先回顾了曲线曲面造型方法的分类以及各自的特点。阐述了CAGD中参数曲线曲面造型的发展历史并介绍了Bzier方法、B样条方法以及非多项式曲线曲面造型方法，后者包括L-样条、螺旋样条、张力样条以及C-曲线等。
文章以Bézier曲线和B样条曲线的特点为基础，在三角函数空间中构造一组具有上述两类曲线特性的三角函数多项式曲线，称其为T-Bézier曲线和T-B样条曲线。它们继承了Bézier曲线和B样条曲线的特点，曲线表示简单、直观。此外由于它们还具有三角函数的优点，故既可以精确表示直线段、二次多项式曲线段又可以精确表示圆弧、椭圆弧等二次曲线以及心脏线、双纽线等超越曲线。特别地，3次均匀T-B样条曲线曲面比同阶均匀B样条(C-B样条)曲线曲面具有更高的光滑度。3次T-Bézier曲线在光滑拼接时也可以达到更高的连续性。最后由于这两类曲线仅由三角函数构成，所以它们较易转化为有理多项式曲线。从而融入到现有的几何造型系统中。
在C-曲线的启示下，本文进一步在T-Bézier曲线及T-B样条曲线中引入控制参数α用以调整曲线形状，构造了另一类自由参数曲线，称其为TC-Bézier曲线及TC-B样条曲线。这两类曲线一方面具有T-Bézier曲线及T-B样条曲线的类似性质和相关二次曲线的精确表示，另一方面由于参数α的引入使得曲线具有更强的表现能力。
文章最后运用同样的方法在双曲函数空间中构造了HC-Bézier曲线。该曲线与TC-Bézier曲线的性质完全类似，此外它既能精确表示直线段又能精确表示双曲线。HC-Bézier曲线中同样具有控制参数α，从而调整曲线形状更加灵活。
对于每一类曲线作者均将它们直接推广到张量积曲面，这些曲面可以精确表示球面、椭球面、双曲面等二次曲面。
This paper summaries the researches on the new schemes of parameter curves and surfaces modeling-curves and surfaces modeling of trigonometric polynomial, which includes curves and surfaces of T-Bezier, T-B-spline, TC-Bezier and TC-B-spline. HC-Bé zier curves and surfaces are also discussed in the space of hyperbolic functions in the end.
After briefly reviewing the classification and respective characters of curves and surfaces modeling, the paper expatiates its history in CAGD. And then we introduce Bezier, B-spline and non-polynomial curves and surfaces modeling, which include L-splines, helix splines, splines in tension and C-curves etc.
By analyzing the characters of Bezier curves and B-spline curves, we construct trigonometric polynomial curves in the space of trigonometric functions, which assume the characters of Bézier curves and B-spline curves. Their representatives are simple and direct. We call them as T-Bezier curves and T-B-spline curves. They not only inherit the advantages of Bezier curves and B-spline curves, also can be used to represent straight lines precisely and some remarkable transcendental curves precisely, such as circular arc, ellipse, cardioids and twisted pair line etc. Especially, the uniform T-B-spline curve of three degrees is smoother than B-spline curve and C-B-spline curve of the same order. When connected smoothly, T-Bezier curve of three degrees can attain more superior continuity. At last, the T-Bézier curves and T-B-spline curves can be converted to rational curves so easily that they can be merged into current geometry modeling rapidly.
Furthermore, with the illuminating of C-curves, the paper construct another freeform parameter curves and surfaces by introducing parameter a in control. We call them as TC-Bezier curves and TC-B-spline curves. On the one hand, they possess the similar characters of T-Bezier curves and T-B-spline curves. On the other hand, by introducing the parameter a, they act the wonderful ability of representation.
At last, we construct hyperbolic polynomial curves in the space of hyperbolic functions. We call them as HC-Bezier curves. Similarly, they not only can represent straight lines precisely, but also can represent quadric curve precisely, such as hyperbola and so on. Parameter a in control is also used in this kind of curve.
The generation of tensor product surfaces of every kind of curves is straightforward; these corresponding tensor product surfaces also contain many special surfaces, including spherical surface, ellipsoid, hyperboloid etc.
引文
[1] 朱心雄，自由曲线曲面造型技术，科学出版社，2000
[2] 苏步青，刘鼎元，计算几何，上海科学技术出版社，1980
[3] 王国瑾，汪国昭，郑建民，计算机辅助几何设计，高等教育出版社，施普林格出版社，2001
[4] Barr, A.H., Global and local Deformation of Solid Primitives, SIGGRAPH'84, ACM Comp.Graph.Vol.18, No.3, 1984,21-30
[5] Sederberg, T.W. and Parry, S.R., Free-Form Deformation of Solid Geometric Models, SIGGRAPH'86, ACM Comp.Graph.Vol.20, No.4, 1986,151-160
[6] Coquillart, S., Extended Free-Form Deformations: A Sculpturing Tool for 3D Geometric Modeling, SIGGRAPH'90, ACM Comp.Graph.Vol.24, No.4, 1990,187-196
[7] Lamousin, H.J. and Waggenspack, W.N., NURBS-Based Free-Form Deformations IEEE Comp. Grap. and Appl. Vol, 11, 1994,59-65
[8] Hoppe, H., DeRose, T. Duchamp, T., Halstead, M., et al., Surface reconstruction from unorganized points, Computer Graphics (SIGGRAPH), 1992, 26:71-78
[9] Muraki, S., Volumetric shape description of range data using "Blobby model", Computer Graphics (SIGGRAPH), 1991, 25:227-235
[10] 吕勇刚，CAGD自由曲线曲面造型中均匀样条的研究，博士学位论文，浙江大学，2002
[11] Farouki, R.T. and Neff, C.A., Analytic properties of plane offset curves, Computer Aided Geometric Design, 1990, 7:83-99
[12] Bloor, M.I.G. and Wilson, M.J., Generating Blend Surfaces Using Partial Differential Equations, Computer Aided Design, 1989, 21:165-171
[13] 施法中，计算机辅助几何设计与非均匀有理B样条，高等教育出版社，2001
[14] Ferguson, J.C., Multivariable curve interpolation, Journal of ACM, 1964,2: 221-228
[15] Coons, S.A., Surfaces for computer aided design of space Figures, Technical Report, MIT, 1964,project MAC-TR-225
[16] Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 1946, 4:45-99
[17] Bzier, P. E., Numerical control-Mathematics and applications, Forrest trans. Wiley, London, 1972
[18] de Boor, C., On calculating with B-splines, Journal of Approximation Theory, 1972, 6:50-62
[19] Piegl, L., Curve fitting algorithm for rough cutting, Computer Aided Design, 1986,18:79-82
[20] Tiller, W., Rational B-splines for curve and surface representation, IEEE Computer Graphics and its Application, 1983, 3:61-69
[21] Piegl, L. and Tiller, W., Curve and Surface construction using B-splines, Computer Aided Design, 1987, 19:487-498
[22] 周蕴时，苏志勋，奚涌江，程少春，CAGD中的曲线与曲面，吉林大学出版社，1993
[23] Schumaker, L. L., Spline functions: Basic Theory, Wiley, New York, 1981,363-499
[24] Pottmann, H., The geometry of Tchebycheffian splines, Computer Aided Geometric Design, 1993, 10:181-210
[25] Pottmann, H. and Wagner, M. G., Helix splines as example of affine Tchebycheffian splines, Advance in Computational Mathematics, 1994, 2:123-142
[26] Koch, P. E. and Lyche, T., Exponential B-splines in tension, in: C. K. Chui, L. L. Schumaker and J. D. Ward, Eds. Approximation Theory Ⅵ,Academic Press, New York, 1989:361-364
[27] Zhang, J. W., C-curves: an extension of cubic curves, Computer Aided Geometric Design, 1996, 13:199-217
[28] Zhang, J. W., C-curves: Two different forms of C-B-Splines, Computer Aided Geometric Design, 1997, 14:31-41
[29] Zhang, J. W., C-Bzier curves and surfaces, Graphical Models and Image Processing, 1999, 61:2-15
[30] Schweikert, D.G., An interpolation curves using a spline in tension, J. Math. Physics, 1966, 45:312-317
[31] Mainar, E., Pea, J.M., Sanchez-Reyes, J., Shape preserving alternatives to the rational Bzier model. Computer Aided Geometric Design, 2001, 18(1): 37-60.
[32] 张帆，康宝生，(n+1)维空间C~n [a，b]上规范B基存在的充要条件，计算机辅助设计与图形学学报，2003，15(5)：566-569。
[33] 陈秦玉，汪国昭，圆弧的C-Bzier曲线表示，软件学报，2002，13(11)：2155-2161
[34] 陈秦玉，杨勋年，汪国昭，四次C-曲线的性质及其应用，高校应用数学学报(A辑)，2003，18(1)：45-50
[35] 樊建华，张纪文，邬义杰，C-Bzier曲线的形状修改，软件学报，2002，13(11)：2194-2199
[36] 金义明，五次C-曲线的生成及其性质，硕士学位论文，浙江大学，2001
[37] Piegl. L, Tiller. W., The NURBS Book(2nd edition), Berlin Heidelberg: Springer-Verlag, 1997.
[38] 唐月红，圆柱螺线的三角有理式Bzier逼近，高等学校计算数学学报，2003，25(1)：31-39
[39] 朱仁芝，程谟嵩，拟合任意空间曲面的三角函数方法，计算机辅助设计与图形学学报，1996，8(2)：108-114
[40] Lyche T, Winther R. Stable recurrence relation for trigonometric B-splines. Journal of Approximation Theory. 1979, 25 (3): 266-279
[41] Han X.L., Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design. 2002, 19(7): 503-512
[42] Jena M.K., Shunmugaraj, P, and Das, P.C., A subdivision algorithm for trigonometric spline curves. Computer Aided Geometric Design. 2002, 19(1): 71-88
[43] Walz, G., Identities for trigonometric B-splines with an application to curve design. BIT 1997, 37 (1): 189-201
[44] Lu Y.G., Wang G.Z., Yang X.N., Uniform trigonometric polynomial B-spline curves SCI CHINA SERF 45 (5): 335-343 OCT 20
[45] Graldine Morin, Analytic Function in Computer Aided Geometric Design. Ph. D. Rice University, 2001.
[46] Cohen E., Lyche T., Riesenfeld R.F., Discrete B-spline and subdivision technique in computer aided geometric design and computer graphics. Computer graphics and image processing, 1980,14
[47] 孙家昶，样条函数与计算几何，科学出版社，1982
[2] 苏步青，刘鼎元，计算几何，上海科学技术出版社，1980
[3] 王国瑾，汪国昭，郑建民，计算机辅助几何设计，高等教育出版社，施普林格出版社，2001
[4] Barr, A.H., Global and local Deformation of Solid Primitives, SIGGRAPH'84, ACM Comp.Graph.Vol.18, No.3, 1984,21-30
[5] Sederberg, T.W. and Parry, S.R., Free-Form Deformation of Solid Geometric Models, SIGGRAPH'86, ACM Comp.Graph.Vol.20, No.4, 1986,151-160
[6] Coquillart, S., Extended Free-Form Deformations: A Sculpturing Tool for 3D Geometric Modeling, SIGGRAPH'90, ACM Comp.Graph.Vol.24, No.4, 1990,187-196
[7] Lamousin, H.J. and Waggenspack, W.N., NURBS-Based Free-Form Deformations IEEE Comp. Grap. and Appl. Vol, 11, 1994,59-65
[8] Hoppe, H., DeRose, T. Duchamp, T., Halstead, M., et al., Surface reconstruction from unorganized points, Computer Graphics (SIGGRAPH), 1992, 26:71-78
[9] Muraki, S., Volumetric shape description of range data using "Blobby model", Computer Graphics (SIGGRAPH), 1991, 25:227-235
[10] 吕勇刚，CAGD自由曲线曲面造型中均匀样条的研究，博士学位论文，浙江大学，2002
[11] Farouki, R.T. and Neff, C.A., Analytic properties of plane offset curves, Computer Aided Geometric Design, 1990, 7:83-99
[12] Bloor, M.I.G. and Wilson, M.J., Generating Blend Surfaces Using Partial Differential Equations, Computer Aided Design, 1989, 21:165-171
[13] 施法中，计算机辅助几何设计与非均匀有理B样条，高等教育出版社，2001
[14] Ferguson, J.C., Multivariable curve interpolation, Journal of ACM, 1964,2: 221-228
[15] Coons, S.A., Surfaces for computer aided design of space Figures, Technical Report, MIT, 1964,project MAC-TR-225
[16] Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 1946, 4:45-99
[17] Bzier, P. E., Numerical control-Mathematics and applications, Forrest trans. Wiley, London, 1972
[18] de Boor, C., On calculating with B-splines, Journal of Approximation Theory, 1972, 6:50-62
[19] Piegl, L., Curve fitting algorithm for rough cutting, Computer Aided Design, 1986,18:79-82
[20] Tiller, W., Rational B-splines for curve and surface representation, IEEE Computer Graphics and its Application, 1983, 3:61-69
[21] Piegl, L. and Tiller, W., Curve and Surface construction using B-splines, Computer Aided Design, 1987, 19:487-498
[22] 周蕴时，苏志勋，奚涌江，程少春，CAGD中的曲线与曲面，吉林大学出版社，1993
[23] Schumaker, L. L., Spline functions: Basic Theory, Wiley, New York, 1981,363-499
[24] Pottmann, H., The geometry of Tchebycheffian splines, Computer Aided Geometric Design, 1993, 10:181-210
[25] Pottmann, H. and Wagner, M. G., Helix splines as example of affine Tchebycheffian splines, Advance in Computational Mathematics, 1994, 2:123-142
[26] Koch, P. E. and Lyche, T., Exponential B-splines in tension, in: C. K. Chui, L. L. Schumaker and J. D. Ward, Eds. Approximation Theory Ⅵ,Academic Press, New York, 1989:361-364
[27] Zhang, J. W., C-curves: an extension of cubic curves, Computer Aided Geometric Design, 1996, 13:199-217
[28] Zhang, J. W., C-curves: Two different forms of C-B-Splines, Computer Aided Geometric Design, 1997, 14:31-41
[29] Zhang, J. W., C-Bzier curves and surfaces, Graphical Models and Image Processing, 1999, 61:2-15
[30] Schweikert, D.G., An interpolation curves using a spline in tension, J. Math. Physics, 1966, 45:312-317
[31] Mainar, E., Pea, J.M., Sanchez-Reyes, J., Shape preserving alternatives to the rational Bzier model. Computer Aided Geometric Design, 2001, 18(1): 37-60.
[32] 张帆，康宝生，(n+1)维空间C~n [a，b]上规范B基存在的充要条件，计算机辅助设计与图形学学报，2003，15(5)：566-569。
[33] 陈秦玉，汪国昭，圆弧的C-Bzier曲线表示，软件学报，2002，13(11)：2155-2161
[34] 陈秦玉，杨勋年，汪国昭，四次C-曲线的性质及其应用，高校应用数学学报(A辑)，2003，18(1)：45-50
[35] 樊建华，张纪文，邬义杰，C-Bzier曲线的形状修改，软件学报，2002，13(11)：2194-2199
[36] 金义明，五次C-曲线的生成及其性质，硕士学位论文，浙江大学，2001
[37] Piegl. L, Tiller. W., The NURBS Book(2nd edition), Berlin Heidelberg: Springer-Verlag, 1997.
[38] 唐月红，圆柱螺线的三角有理式Bzier逼近，高等学校计算数学学报，2003，25(1)：31-39
[39] 朱仁芝，程谟嵩，拟合任意空间曲面的三角函数方法，计算机辅助设计与图形学学报，1996，8(2)：108-114
[40] Lyche T, Winther R. Stable recurrence relation for trigonometric B-splines. Journal of Approximation Theory. 1979, 25 (3): 266-279
[41] Han X.L., Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design. 2002, 19(7): 503-512
[42] Jena M.K., Shunmugaraj, P, and Das, P.C., A subdivision algorithm for trigonometric spline curves. Computer Aided Geometric Design. 2002, 19(1): 71-88
[43] Walz, G., Identities for trigonometric B-splines with an application to curve design. BIT 1997, 37 (1): 189-201
[44] Lu Y.G., Wang G.Z., Yang X.N., Uniform trigonometric polynomial B-spline curves SCI CHINA SERF 45 (5): 335-343 OCT 20
[45] Graldine Morin, Analytic Function in Computer Aided Geometric Design. Ph. D. Rice University, 2001.
[46] Cohen E., Lyche T., Riesenfeld R.F., Discrete B-spline and subdivision technique in computer aided geometric design and computer graphics. Computer graphics and image processing, 1980,14
[47] 孙家昶，样条函数与计算几何，科学出版社，1982