光滑曲面上曲线的生成及其性质研究
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摘要
本文研究光滑曲面上曲线的计算机表示、插值与逼近等。具体讨论圆柱螺旋线的三角有理Bézier表示和球面NURBS曲线的生成。
首先，归纳总结极坐标有理Bézier方法和B样条方法以及它们的性质。还给出B-样条曲线的de Boor算法和节点插入算法。
其次，给出圆锥曲线弧及其补弧的三角有理Bézier表示；以及任意角度二次曲线弧的极坐标B样条表示。
再次，建立柱坐标系，给出圆柱螺线的三角有理Bézier表示。对于其控制顶点的确定给出了两种方法。方法一，使逼近螺线在连接点处GC~2连续，得到的逼近螺线与理论螺线在两端点及中心理误差为零。方法二，增加重合点，使理论螺线和逼近螺线在五点处重合，从而反求出控制顶点，使误差带更窄。两种方法均给出误差估计，使逼近螺线可达到任何预先给定的精确阶。
最后，给出球面NURBS曲线生成算法：用球面上测地线——劣大圆弧代替直线段，将欧氏空间R~3中的de Boor递推算法推广到球面上构造曲线。并且讨论了这种曲线的若干性质，有类似于欧氏空间中的性质，还指出其不具有类似于欧氏空间中的NURBS曲线的分裂性质，同时给出球面NURBS曲线的插入节点算法。还给出球面上等距节点二次和三次B样条曲线的插值方法。作为对曲线生成算法和性质以及插值方法的应用，最后给出了一些图形实例。
The interpolation, approximation and other things for curves on smooth surfaces with computer are studied. The triangular rational Bezier representation of cylindrical helix and the generation of the NURBS curves on sphere are discussed in detail.
Firstly, the thesis devotes to induct and develop the methods and properties for the rational triangular Bezier and B-splines in polar coordinates system. The de Boor algorithm and a knot insertion algorithm are also given for B-spline curves.
Secondly, the various conic arcs and conies are generated by the rational triangular Bezier and the B-spline method in polar coordinates system.
Thirdly, the spatial cylindrical helix is approximated by rational triangular Bezier curve in cylindrical coordinate system. Two methods are given to get the control vertexes, by making the two approximated helix segments GC2 continuous at their connecting point and leting the theoretical helix and the approximated one coincident at five points respectively. The theory error of one approximated helix equals zero at the two end -points and the center and the other's equals zero at five points. How the parameters of above two helixes affect their shapes is discussed. Furthermore, error analysis and distribution are given. Any prescribed precision can be arrived by the approximated helixes.
Finally, a method is put forward to construct the NURBS curves on sphere, which extends the de Boor recursive algorithm in R3 to one on the sphere by replacing the geodesic distances for the lines and studies their many geometric properties analogous to those in Euclidean spaces, such as the differential property, the local property, the parameter invariance under a projective transformation, and so on. It is also pointed out that these curves are devoid of split property possessed by NURBS curves in Euclidean spaces. At the same time, a knot insertion algorithm is also given for the NURBS curves on sphere, then interpolation for curves on sphere is presented by spherical quadratic and cubic uniform B-spline. However, the quadratic curve is only C?continuous, while the cubic's is C1 continuous. As an application of the algorithm and the properties, some examples are given.
首先，归纳总结极坐标有理Bézier方法和B样条方法以及它们的性质。还给出B-样条曲线的de Boor算法和节点插入算法。
其次，给出圆锥曲线弧及其补弧的三角有理Bézier表示；以及任意角度二次曲线弧的极坐标B样条表示。
再次，建立柱坐标系，给出圆柱螺线的三角有理Bézier表示。对于其控制顶点的确定给出了两种方法。方法一，使逼近螺线在连接点处GC~2连续，得到的逼近螺线与理论螺线在两端点及中心理误差为零。方法二，增加重合点，使理论螺线和逼近螺线在五点处重合，从而反求出控制顶点，使误差带更窄。两种方法均给出误差估计，使逼近螺线可达到任何预先给定的精确阶。
最后，给出球面NURBS曲线生成算法：用球面上测地线——劣大圆弧代替直线段，将欧氏空间R~3中的de Boor递推算法推广到球面上构造曲线。并且讨论了这种曲线的若干性质，有类似于欧氏空间中的性质，还指出其不具有类似于欧氏空间中的NURBS曲线的分裂性质，同时给出球面NURBS曲线的插入节点算法。还给出球面上等距节点二次和三次B样条曲线的插值方法。作为对曲线生成算法和性质以及插值方法的应用，最后给出了一些图形实例。
The interpolation, approximation and other things for curves on smooth surfaces with computer are studied. The triangular rational Bezier representation of cylindrical helix and the generation of the NURBS curves on sphere are discussed in detail.
Firstly, the thesis devotes to induct and develop the methods and properties for the rational triangular Bezier and B-splines in polar coordinates system. The de Boor algorithm and a knot insertion algorithm are also given for B-spline curves.
Secondly, the various conic arcs and conies are generated by the rational triangular Bezier and the B-spline method in polar coordinates system.
Thirdly, the spatial cylindrical helix is approximated by rational triangular Bezier curve in cylindrical coordinate system. Two methods are given to get the control vertexes, by making the two approximated helix segments GC2 continuous at their connecting point and leting the theoretical helix and the approximated one coincident at five points respectively. The theory error of one approximated helix equals zero at the two end -points and the center and the other's equals zero at five points. How the parameters of above two helixes affect their shapes is discussed. Furthermore, error analysis and distribution are given. Any prescribed precision can be arrived by the approximated helixes.
Finally, a method is put forward to construct the NURBS curves on sphere, which extends the de Boor recursive algorithm in R3 to one on the sphere by replacing the geodesic distances for the lines and studies their many geometric properties analogous to those in Euclidean spaces, such as the differential property, the local property, the parameter invariance under a projective transformation, and so on. It is also pointed out that these curves are devoid of split property possessed by NURBS curves in Euclidean spaces. At the same time, a knot insertion algorithm is also given for the NURBS curves on sphere, then interpolation for curves on sphere is presented by spherical quadratic and cubic uniform B-spline. However, the quadratic curve is only C?continuous, while the cubic's is C1 continuous. As an application of the algorithm and the properties, some examples are given.
引文
[1] Barnhill R.E. & Riesenfeld R.F., CAGD, Academic press, New York, 1974.
[2] 苏步青，刘鼎元，“计算几何”，上海科技出版社，1981
[3] 苏步青，“计算几何的兴起”，自然杂志，1987，No．7．
[4] Coons S., Surfaces for computer aided design of space figures, AD 663504, 1967
[5] Bamhill R. E., Riesenfeld R. F., Computer-aided geometric design, academic press, 1974
[6] Ferguson J.C., Multivariable curve Interpolation, report No. D2-22504, The Boeing Co.,Seattle, Washington, 1963.
[7] Ferguson J.C., Multivariable Curve Interpolation, J.ACM, Vol. 11, No. 2, 1964, 221-228
[8] Bézier P., Numerical Control-Mathematics and Applications (Translated by A.R. Forrest), John Wiley and Sons, London, 1972.
[9] Bézier P., Example of an Existing system in the Motor Industry: The NUISURF System, Proc. Roy. Soc. of London, Vol. a321, 1971, 207-218.
[10] Riesenfeld R.F., Applications of B-Spline Approximation to Geometric Problem of CAD, Ph.D Thesis, Syracuse University,1973.
[11] Gordon W.J. & Riesenfeld R.F., B-Spline Curves and surfaces in CAGD, Barnhill R.E. & Riesenfeld R.F., Editors, Academic Press, 1974.
[12] Hartley P.J. & Judd, Parameterization of Bézier-Type B-Spline Curves and Surfaces, CAD, Vol. 10, No. 2, 1978, 130-134.
[13] Piegl L. & Tiller W., Curve and Surface Constructions Using Rational B-Splines, CAD, Vol. 19, No. 9, 1987.
[14] Piegl L. & Tiller W., The NURBS Book, Springer, New York, 1995.
[15] Verspprille K., Computer aided design applications of the rational B-Spline Approximation form, Ph. D. Dissertation, Syracuse University, 1975.
[16] Sánchez-Reyes J., Single-Valued Spline Curves in Polar Coordinates, CAD, Vol. 22,No. 1, 1990, 19-26.
[17] Sánchez-Reyes J., Single-Valued Spline Curves in Polar Coordinates, CAD, Vol. 24, No. 31, 1992, 307-315.
[18] Sánchez-Reyes J., Higher-Order Bézier Circles, CAD, Vol. 29, No. 6, 1997, 469-472.
[19] Sánchez-Reyes J., Bézier-Representation of Epitrochoids and Hypotro-choids, CAD, No. 31, 1999, 747-750.
[20] Piegl L., Representation of Quadric Primitives by Rational Polynomials, Computer Aided Geometric Design, No.2, 1985, 151-156.
[21] 施法中，“各种角度圆弧的二次NURBS表示”，计算机辅助设计与图形学学报，Vol．5，No．4，1993,247-251．
[22] 秦开怀，关右江，“圆弧曲线的三次NURBS表示”，计算机学报，Vol．18，No．2，1995，146-152．
[23] Piegl L. & Tiller W., A Menagerie of Ratonal B-Spline Circles, IEEE CG&A, Vol.9, No.5, 1989, 48-56.
[24] 范劲松，安军，徐宗俊，“用三次NURBS表示圆弧与整圆的算法研究”，计算机辅助设计与图形学学报，Vol．9，No．5，1997，391-395．
[25] Chou J. J., Higher Oder Bézier Circles, CAD, Vol.27, NO.4, 1995, 303-309.
[26] 施法中，“计算机辅助几何设计与非均匀有理B样条(CAGD&NURBS))”，北京航空航天大学出版社，1994
[27] Sanchez-Reyes J., Single-valued Surfaces in Spherical Coordinates, Computer-Aided Geometric design, vol. 11, No. 5, 1994, 491-517.
[28] Mick S. & Rōschel o., Interpolation of Helical Patches by Kinematic Rational Bézier Patches, Comput. & Graph., Vol. 14, No. 2, 1990, 275-280.
[29] Imre Juhász, Approximating the Helix With Rational Polynomials, Computer Aided Geometric Design, No. 2, 1985, 151-155.
[30] 孙越泓，“计算几何中的空间圆柱螺线”，硕士学士论文，南京航空航天大学，2000年．
[31] 孙家昶，郑全琳，“曲线的圆弧逼近与双圆弧逼近”，计算数学，Vol．3，No．2，1981，97-112．
[32] 董光昌，梁友栋，何援军，“样条曲线拟合与双圆弧逼近”，应用数学学报，Vol．1，No．4，1978,330-340．
[33] Meek D.S., Walton D. J., Approxmafion of discrete data by G~1 arc splines, Computer Aided Design, Vol.24, No.6, 1992,301-306.
[34] Ahn Y. J., Kim H.O., Lee, K. Y., G~1 arc spline approximation of quadratic Bézier curves, Computer Aided Design, Vol.30, No.8, 1998, 615-620.
[35] Ong C. J., Wong Y. S., Loh H. T., Hong X. G., An optimization approach for biarc curve-fitting of B-spline curves, Computer Aided Design, Vol.28, No. 12, 1996, 951-959.
[36] Mehlum E., Nonlinear splines, In: Barnhill R. E. and Riesenfeld R. F., eds., Computer Aided Geometric Design, Academic press, New York, 1974, 173-207.
[37] Shoemake K., Animation rotation with quaternion curves, SIGGRAPH, Vol. 19 No.3,1985, 245-254.
[38] 张怀，“球面Bézier曲线的生成及其性质”，学位论文，复旦大学，1994．
[39] Farouki R. T., the Conformal Mao Z-Z2 of the Hodograph Planes, Computer-Aided Geometric Design, Vol. 11, No. 4,1994,363-390．
[40] Lee E T Y, The Rational Bezier Representation for Conies, Geometric Modelling: Algorithms and New Trends SIAM, 1987．
[41] Piegl L., On NURBS : a survey, IEEE Comput. Graph. Appl. Vol.11, No.l, 1991,55-71．
[42] 李庆扬，易大义，王以超，“现代数值分析”，高等教育出版社, 1995．
[43] De Boor C., A practical guides to splines, Verlag: Springer, 1978．
[44] Lawson C. L., C2 surface interpolation for scattered data on a sphere, Rocky Mountain Jourmal of Mathematics, Vol.14, No.l, 1984,177-202．
[2] 苏步青，刘鼎元，“计算几何”，上海科技出版社，1981
[3] 苏步青，“计算几何的兴起”，自然杂志，1987，No．7．
[4] Coons S., Surfaces for computer aided design of space figures, AD 663504, 1967
[5] Bamhill R. E., Riesenfeld R. F., Computer-aided geometric design, academic press, 1974
[6] Ferguson J.C., Multivariable curve Interpolation, report No. D2-22504, The Boeing Co.,Seattle, Washington, 1963.
[7] Ferguson J.C., Multivariable Curve Interpolation, J.ACM, Vol. 11, No. 2, 1964, 221-228
[8] Bézier P., Numerical Control-Mathematics and Applications (Translated by A.R. Forrest), John Wiley and Sons, London, 1972.
[9] Bézier P., Example of an Existing system in the Motor Industry: The NUISURF System, Proc. Roy. Soc. of London, Vol. a321, 1971, 207-218.
[10] Riesenfeld R.F., Applications of B-Spline Approximation to Geometric Problem of CAD, Ph.D Thesis, Syracuse University,1973.
[11] Gordon W.J. & Riesenfeld R.F., B-Spline Curves and surfaces in CAGD, Barnhill R.E. & Riesenfeld R.F., Editors, Academic Press, 1974.
[12] Hartley P.J. & Judd, Parameterization of Bézier-Type B-Spline Curves and Surfaces, CAD, Vol. 10, No. 2, 1978, 130-134.
[13] Piegl L. & Tiller W., Curve and Surface Constructions Using Rational B-Splines, CAD, Vol. 19, No. 9, 1987.
[14] Piegl L. & Tiller W., The NURBS Book, Springer, New York, 1995.
[15] Verspprille K., Computer aided design applications of the rational B-Spline Approximation form, Ph. D. Dissertation, Syracuse University, 1975.
[16] Sánchez-Reyes J., Single-Valued Spline Curves in Polar Coordinates, CAD, Vol. 22,No. 1, 1990, 19-26.
[17] Sánchez-Reyes J., Single-Valued Spline Curves in Polar Coordinates, CAD, Vol. 24, No. 31, 1992, 307-315.
[18] Sánchez-Reyes J., Higher-Order Bézier Circles, CAD, Vol. 29, No. 6, 1997, 469-472.
[19] Sánchez-Reyes J., Bézier-Representation of Epitrochoids and Hypotro-choids, CAD, No. 31, 1999, 747-750.
[20] Piegl L., Representation of Quadric Primitives by Rational Polynomials, Computer Aided Geometric Design, No.2, 1985, 151-156.
[21] 施法中，“各种角度圆弧的二次NURBS表示”，计算机辅助设计与图形学学报，Vol．5，No．4，1993,247-251．
[22] 秦开怀，关右江，“圆弧曲线的三次NURBS表示”，计算机学报，Vol．18，No．2，1995，146-152．
[23] Piegl L. & Tiller W., A Menagerie of Ratonal B-Spline Circles, IEEE CG&A, Vol.9, No.5, 1989, 48-56.
[24] 范劲松，安军，徐宗俊，“用三次NURBS表示圆弧与整圆的算法研究”，计算机辅助设计与图形学学报，Vol．9，No．5，1997，391-395．
[25] Chou J. J., Higher Oder Bézier Circles, CAD, Vol.27, NO.4, 1995, 303-309.
[26] 施法中，“计算机辅助几何设计与非均匀有理B样条(CAGD&NURBS))”，北京航空航天大学出版社，1994
[27] Sanchez-Reyes J., Single-valued Surfaces in Spherical Coordinates, Computer-Aided Geometric design, vol. 11, No. 5, 1994, 491-517.
[28] Mick S. & Rōschel o., Interpolation of Helical Patches by Kinematic Rational Bézier Patches, Comput. & Graph., Vol. 14, No. 2, 1990, 275-280.
[29] Imre Juhász, Approximating the Helix With Rational Polynomials, Computer Aided Geometric Design, No. 2, 1985, 151-155.
[30] 孙越泓，“计算几何中的空间圆柱螺线”，硕士学士论文，南京航空航天大学，2000年．
[31] 孙家昶，郑全琳，“曲线的圆弧逼近与双圆弧逼近”，计算数学，Vol．3，No．2，1981，97-112．
[32] 董光昌，梁友栋，何援军，“样条曲线拟合与双圆弧逼近”，应用数学学报，Vol．1，No．4，1978,330-340．
[33] Meek D.S., Walton D. J., Approxmafion of discrete data by G~1 arc splines, Computer Aided Design, Vol.24, No.6, 1992,301-306.
[34] Ahn Y. J., Kim H.O., Lee, K. Y., G~1 arc spline approximation of quadratic Bézier curves, Computer Aided Design, Vol.30, No.8, 1998, 615-620.
[35] Ong C. J., Wong Y. S., Loh H. T., Hong X. G., An optimization approach for biarc curve-fitting of B-spline curves, Computer Aided Design, Vol.28, No. 12, 1996, 951-959.
[36] Mehlum E., Nonlinear splines, In: Barnhill R. E. and Riesenfeld R. F., eds., Computer Aided Geometric Design, Academic press, New York, 1974, 173-207.
[37] Shoemake K., Animation rotation with quaternion curves, SIGGRAPH, Vol. 19 No.3,1985, 245-254.
[38] 张怀，“球面Bézier曲线的生成及其性质”，学位论文，复旦大学，1994．
[39] Farouki R. T., the Conformal Mao Z-Z2 of the Hodograph Planes, Computer-Aided Geometric Design, Vol. 11, No. 4,1994,363-390．
[40] Lee E T Y, The Rational Bezier Representation for Conies, Geometric Modelling: Algorithms and New Trends SIAM, 1987．
[41] Piegl L., On NURBS : a survey, IEEE Comput. Graph. Appl. Vol.11, No.l, 1991,55-71．
[42] 李庆扬，易大义，王以超，“现代数值分析”，高等教育出版社, 1995．
[43] De Boor C., A practical guides to splines, Verlag: Springer, 1978．
[44] Lawson C. L., C2 surface interpolation for scattered data on a sphere, Rocky Mountain Jourmal of Mathematics, Vol.14, No.l, 1984,177-202．