隐式代数曲线在CAGD中的性质及应用研究
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摘要
在计算机辅助几何设计(Computer aided geometrid design)领域中，熟知有两种定义曲线曲面的方法，参数形式及隐式形式。参数形式以其构造简单，计算容易等特点而流行于世并成为几何设计的主流，然而近20年的研究与使用经验表明隐式形式也有参数形式无法比拟的优点，本文仅就隐式曲线在计算机辅助几何设计中应用及性质进行了分析和研究。其主要结果如下：
本文首先对隐式曲线研究的现状、主流和趋势等作了总的概述。在此基础上对近20年来隐式曲线在计算机辅助几何设计中的应用及各种隐式形式的曲线特点进行了总结和评价。
本文在对插值一类几何约束的隐式代数曲线的构造基础上，给出了这样的隐式三次代数曲线二阶几何连续光滑拼接的条件，并给出了实验结果。理论分析及实验结果表明这样的拼接仍有一个相应的自由度，能对曲线形状进行调节，以达到较好的设计效果。
根据G~(n-1)函数样条(G~(n-1)-functional spline)曲线的思想，本文对正方形四顶点采用三次隐式代数曲线进行插值构造出二阶几何连续光滑封闭曲线，利用这种方法，通过适当地选择基曲线，可以在不引入任何控制点的情况下，使设计出的三次隐式曲线在插值节点处具有要求的曲率，并且仍然有一个自由参数可以对曲线形状进行调整，因此具有较好的性质。
本文在参—参曲面交线基础上，详细推导了隐—隐形式两曲面空间交线的微分几何性质，这些性质包括截交及切交下的切矢，曲率矢以及高阶导矢等。这些性质的给出将有利于空间隐式曲线在造型中的进一步研究。
In Computer-Aided Geometric Design field, two forms defined curves/surfaces are well known: parameter form and implicit one. Parameter form becomes mainstream of geometric design because of characteristic such as: simple construction, easy computation, etc.. While near 20 years research illustrates implicit form has advantage that parameter one hasn't. In this paper only the application and properties of implicit curves in CAGD have been researched. The main results are as follows:
Firstly the researching status quo, mainstream and tend of implicit curves are summarized in this paper. Based on these near 20 years the applicant of implicit curves in CAGD and the characteristics of all sorts of implicit one are summarized and evaluated.
In this paper the G2 -continuous blending conditions of two algebraic curves implicitly defined based on the construction of algebraic curves that interpolate the geometric constraints are presented. The theoretical analysis and the experimental results demonstrate that cubic algebraic curves having G2-continuous blending still have a freedom to fit the shape of the curve.
G2-continuous closed curves are constructed through interpolating the vertices of a square using cubic implicit algebraic curves by Gn-1-functional spline idea in this paper. In virtue of this ways, the cubic curves implicitly defined are constructed and a freedom is still had to adjust the curve shapes without introducing any reference points. The experimental results also illustrate the good design effect of this approach.
Based on parameter-parameter surfaces intersection curve, the differential geometric properties of implicit-implicit ones are in detail in the paper. These properties include tangents, curvature vector and higher-order derivative vector in cases of transversal intersection and tangent intersection. These results will be useful to the next research of space implicit curve in geometric modeling.
本文首先对隐式曲线研究的现状、主流和趋势等作了总的概述。在此基础上对近20年来隐式曲线在计算机辅助几何设计中的应用及各种隐式形式的曲线特点进行了总结和评价。
本文在对插值一类几何约束的隐式代数曲线的构造基础上，给出了这样的隐式三次代数曲线二阶几何连续光滑拼接的条件，并给出了实验结果。理论分析及实验结果表明这样的拼接仍有一个相应的自由度，能对曲线形状进行调节，以达到较好的设计效果。
根据G~(n-1)函数样条(G~(n-1)-functional spline)曲线的思想，本文对正方形四顶点采用三次隐式代数曲线进行插值构造出二阶几何连续光滑封闭曲线，利用这种方法，通过适当地选择基曲线，可以在不引入任何控制点的情况下，使设计出的三次隐式曲线在插值节点处具有要求的曲率，并且仍然有一个自由参数可以对曲线形状进行调整，因此具有较好的性质。
本文在参—参曲面交线基础上，详细推导了隐—隐形式两曲面空间交线的微分几何性质，这些性质包括截交及切交下的切矢，曲率矢以及高阶导矢等。这些性质的给出将有利于空间隐式曲线在造型中的进一步研究。
In Computer-Aided Geometric Design field, two forms defined curves/surfaces are well known: parameter form and implicit one. Parameter form becomes mainstream of geometric design because of characteristic such as: simple construction, easy computation, etc.. While near 20 years research illustrates implicit form has advantage that parameter one hasn't. In this paper only the application and properties of implicit curves in CAGD have been researched. The main results are as follows:
Firstly the researching status quo, mainstream and tend of implicit curves are summarized in this paper. Based on these near 20 years the applicant of implicit curves in CAGD and the characteristics of all sorts of implicit one are summarized and evaluated.
In this paper the G2 -continuous blending conditions of two algebraic curves implicitly defined based on the construction of algebraic curves that interpolate the geometric constraints are presented. The theoretical analysis and the experimental results demonstrate that cubic algebraic curves having G2-continuous blending still have a freedom to fit the shape of the curve.
G2-continuous closed curves are constructed through interpolating the vertices of a square using cubic implicit algebraic curves by Gn-1-functional spline idea in this paper. In virtue of this ways, the cubic curves implicitly defined are constructed and a freedom is still had to adjust the curve shapes without introducing any reference points. The experimental results also illustrate the good design effect of this approach.
Based on parameter-parameter surfaces intersection curve, the differential geometric properties of implicit-implicit ones are in detail in the paper. These properties include tangents, curvature vector and higher-order derivative vector in cases of transversal intersection and tangent intersection. These results will be useful to the next research of space implicit curve in geometric modeling.
引文
[1] Patterson R. R. (1988) Parametric curves as algebraic curves, Computer Aided Geometric Design 5: 139—159
[2] Qu Yongming, Sun Shouqian, Chert Falai(1997) Approximate parametrization of algebraic curves, J. China University of Sci. and Tec. 27: 382—387(in Chinese) (屈永明，孙建忠，陈发来，1997，代数曲线的近似参数化，中国科学技术大学学报27：382—387)
[3] Waggenspack W. N. and Anderson D. C. (1989) Piecewise approximation to algebraic curves, Computer Aided Geometric Design 6: 33—53
[4] Zhang Sanyuan, Bao Hujun, Wei Baogang(2001) Cubic algebraic curves based on geometric constraints, Computer Aided Geometric Design 18: 299—307
[5] Zhang Sanyuan, Sun Shouqian, Pan Yunhe(2001)Cubic algebraic curves interpolation with geometric constraints, Chinese J. Computers 24: 509—515(in Chinese)(张三元，孙守谦，潘云鹤，2001，基于几何约束的三次代数曲线插值，计算机学报24：509—515)
[6] Zhang sanyuan(2000) Modeling G~2 continuity curves based algebraic curve segment, Chinese J. Computers 23: 153—157(in Chinese)(张三元，2000，基于代数曲线段的G~2连续的代数曲线的造型方法，计算机学报23：153—157)
[7] Li J., Hosckek J., Hartman E.. (1990) G~(n-1)-function splines for interpolation and approximation of curves, surfaces and solids, Computer Aided Geometric Design 7: 209—220
[8] Hartman E. (1990) Blending of implicit surfaces with function splines, Computer Aided Design 22: 500—506
[9] Zhang Sanyuan, Liang Youdong(1999) Global modeling for G~1 tube-like surface, J. CAD&CG 11: 4—7(in Chinese)(张三元，梁友栋，1999，G~1管状曲面的整体造型方法，计算机辅助设计与图形学学报11：4—7)
[10] Sederberg, T. (1984) Planar piecewise algebraic curves, Computer Aided Geometric Design 1: 241—255
[11] Bajaj, C., Chen J., Holt R., Netravali A. (1999a) Energy formulations of A-splines, Computer Aided Geometric Design 16: 39—59
[12] Bajaj, C., Xu G. (1999b) A-splines: local interpolation and approximation using
C~k-continuity piecewise real algebraic curves, Computer Aided Geometric Design 16: 557—578
[13] Paluszny, M., Patterson, R. R. (1998) Geometric control of G~2 -cubic A-splines, Computer Aided Geometric Design 13: 261—287
[14] Xu G., Bajaj C., Xue W. (2000a) Regular algebraic curve segment(Ⅰ)—definition and characteristics, Computer Aided Geometric Design 17: 485-501
[15] Xu G., Bajaj C., Chu C. (2000b) Regular algebraic curve segment(Ⅱ)—interpolation and approximation, Computer Aided Geometric Design 17: 503—519
[16] Bajaj C., Xu G. (2001) Regular algebraic curve segment(Ⅲ)—application in interactive design and data fitting, Computer Aided Geometric Design 18: 149—173
[17] Fletcher Y, Mcallister F D. Automatic tension adjustment for interpolatory splines IEEE Computer Graphics & Appliment, 1990, 10(1): 10—17
[18] Shi F..《CAGD&NURBS》 Beijing: Beijing University of Aeronautics and Astronautics Press. 1994(in Chinese)(施法中《计算机辅助几何设计与非均匀有理B样条》北京：北京航空航天大学出版社，1994)
[19] Fang K, Tan J, Zhu G. An algorithm for constructing C~2 shape-preserving of monotonicity-preserving interpolating polynomial shaping. Journal of Computer-Aided Design & Computer Graphics, 1996, 8(5): 361—366(in Chinese)(方逵，谭建荣，朱国庆 C~2保单调或保形的插值多项式样条算法 计算机辅助设计及图形学学报 1996 8(5)：361—366)
[20] Brodlie W K, Butt S. Preserving convexity using piecewise cubic interpolation Computer & Graphics, 1991 15(1)15—23
[21] Zhang S. The geometric invariant of implicit curve and surface and geometric continuity between implicit surfaces. Chinese J. Computers, 1999, 22(7): 774—776(in Chinese)(张三元 基于代数曲线段的几何不变量及几何连续性 计算机学报 1999 22(7)774—776)
[22] Fletcher Y, Mcallister F D. Automatic tension adjustment for interpolatory splines IEEE Computer Graphics & Appliment, 1990, 10(1): 10—17
[23] Brodlie W K, Butt S. Preserving convexity using piecewise cubic interpolation Computer & Graphics, 1991 15(1)15—23
[24] Xiuzi Ye, Takashi Maekawa(1999) Differential geometry of intersection curves of two
surfaces Computer Aided Geometric Design 16: 767—788
[25] 石雄辉(2002) 高阶几何连续过渡曲面的造型研究与应用 西北工业大学硕士论文
[26] 梅向明，黄敬之《微分几何》高等教育出版社(第2版) 1988
[27] 梁友栋 曲线、曲面几何连续性问题 数学年刊 1990，11A：3，374-386
[28] Tieru Wu, Yunshi Zhou(2000) On blending of several quadratic algebraic surfaces, Computer Aided Geometric Design 17: 759—766
[29] Makus Schichtel(1993), G~2 blend surfaces and filling of N-sided holes, IEEE Computer Graphics & Appliment, sep. 68—73
[30] Erich Hartmann(1995), Blending an implict with a parametric surface, Computer Aided Geometric Design 12: 825—835
[31] 徐国良(1997)，CAGD中的隐式曲线与曲面，数值计算与计算机应用，199，7(6)：114—124
[32] Hans Hagen, etc.(1996), Surfaces design using triangle patches, Computer Aided Geometric Design 13: 895—904
[33] Guoliang Xu, etc.(2001), C~1 modeling with A-patch from rational trivariate functions, Computer Aided Geometric Design 18: 221—243
[34] Martin R., Shou H., Voiculescu I., Bowyer A., Wang G. (2002) Comparison of interval methods for plotting algebraic curves, CAGD 19: 553—587
[35] Chandler R. E.(1988) A tracking algorithm for implicitly defined curves, IEEE Computer Graphics Appl. 8: 83—89
[36] Duff T. (1992) Interval arithmetic and recursive subdivision for implicit functions and constructive solid geometry, Computer Graphics 26: 131—138
[37] Gopalsamy S., Khandekar D., Mudur S.P.(1991) A new method of valuating compact geometric bounds for use in subdivision algorithms, CAGD 8: 337—356
[38] Hu C. Y., Patrikalakis N. M., Ye X. (1996a) Robust interval solid modeling, Part Ⅰ: Representation, CAD 28: 807—817
[39] Hu C. Y., Patrikalakis N. M., Ye X. (1996b) Robust interval solid modeling, Part Ⅱ: Boundary evaluation, CAD 28: 819—830
[40] Shou H., Martin R., Voiculescu I., Bowyer A., Wang G. (2002) Affine arithmetic in matrix form for algebraic curve drawing, Progress in Natural Science(China)12: 77—80
[41] Snyder J. M. (1992) Interval analysis for computer graphics, Computer Graphics 26: 121—130
[42] Taubin G. (1994) Distance approximation for rasterizing implicit curves, ACM Trans. on
Graphics 13: 3—42
[43] Sederberg T. W., Chen F. (1995) Implicitization using moving curves and surfaces, in: Proc. SIGGRAPH, pp, 301—308
[44] Marco A. & Martinez J.J. (2001) Using polynomial interpolation for implicitizing algebraic curves, CAGD 18: 309—319
[45] Marco A. & Martinez J.J. (2002) Implicitization of rational surfaces by means of polynomial interpolation, CAGD 19: 327—344
[46] Patterson R.R. (2002) Using duality to implieitize and find cusps and inflection point of Bezier curves, CAGD 19: 433—444
[47] Berry T.G.& Patterson R.R. (2001) Implicitization and parametrization of nonsingular cubic surfaces, CAGD 18: 723—738
[48] Chen F. & Sederberg T. (2002) A new implicit representation of a planar curve with high order singularity, CAGD 19: 151—167
[2] Qu Yongming, Sun Shouqian, Chert Falai(1997) Approximate parametrization of algebraic curves, J. China University of Sci. and Tec. 27: 382—387(in Chinese) (屈永明，孙建忠，陈发来，1997，代数曲线的近似参数化，中国科学技术大学学报27：382—387)
[3] Waggenspack W. N. and Anderson D. C. (1989) Piecewise approximation to algebraic curves, Computer Aided Geometric Design 6: 33—53
[4] Zhang Sanyuan, Bao Hujun, Wei Baogang(2001) Cubic algebraic curves based on geometric constraints, Computer Aided Geometric Design 18: 299—307
[5] Zhang Sanyuan, Sun Shouqian, Pan Yunhe(2001)Cubic algebraic curves interpolation with geometric constraints, Chinese J. Computers 24: 509—515(in Chinese)(张三元，孙守谦，潘云鹤，2001，基于几何约束的三次代数曲线插值，计算机学报24：509—515)
[6] Zhang sanyuan(2000) Modeling G~2 continuity curves based algebraic curve segment, Chinese J. Computers 23: 153—157(in Chinese)(张三元，2000，基于代数曲线段的G~2连续的代数曲线的造型方法，计算机学报23：153—157)
[7] Li J., Hosckek J., Hartman E.. (1990) G~(n-1)-function splines for interpolation and approximation of curves, surfaces and solids, Computer Aided Geometric Design 7: 209—220
[8] Hartman E. (1990) Blending of implicit surfaces with function splines, Computer Aided Design 22: 500—506
[9] Zhang Sanyuan, Liang Youdong(1999) Global modeling for G~1 tube-like surface, J. CAD&CG 11: 4—7(in Chinese)(张三元，梁友栋，1999，G~1管状曲面的整体造型方法，计算机辅助设计与图形学学报11：4—7)
[10] Sederberg, T. (1984) Planar piecewise algebraic curves, Computer Aided Geometric Design 1: 241—255
[11] Bajaj, C., Chen J., Holt R., Netravali A. (1999a) Energy formulations of A-splines, Computer Aided Geometric Design 16: 39—59
[12] Bajaj, C., Xu G. (1999b) A-splines: local interpolation and approximation using
C~k-continuity piecewise real algebraic curves, Computer Aided Geometric Design 16: 557—578
[13] Paluszny, M., Patterson, R. R. (1998) Geometric control of G~2 -cubic A-splines, Computer Aided Geometric Design 13: 261—287
[14] Xu G., Bajaj C., Xue W. (2000a) Regular algebraic curve segment(Ⅰ)—definition and characteristics, Computer Aided Geometric Design 17: 485-501
[15] Xu G., Bajaj C., Chu C. (2000b) Regular algebraic curve segment(Ⅱ)—interpolation and approximation, Computer Aided Geometric Design 17: 503—519
[16] Bajaj C., Xu G. (2001) Regular algebraic curve segment(Ⅲ)—application in interactive design and data fitting, Computer Aided Geometric Design 18: 149—173
[17] Fletcher Y, Mcallister F D. Automatic tension adjustment for interpolatory splines IEEE Computer Graphics & Appliment, 1990, 10(1): 10—17
[18] Shi F..《CAGD&NURBS》 Beijing: Beijing University of Aeronautics and Astronautics Press. 1994(in Chinese)(施法中《计算机辅助几何设计与非均匀有理B样条》北京：北京航空航天大学出版社，1994)
[19] Fang K, Tan J, Zhu G. An algorithm for constructing C~2 shape-preserving of monotonicity-preserving interpolating polynomial shaping. Journal of Computer-Aided Design & Computer Graphics, 1996, 8(5): 361—366(in Chinese)(方逵，谭建荣，朱国庆 C~2保单调或保形的插值多项式样条算法 计算机辅助设计及图形学学报 1996 8(5)：361—366)
[20] Brodlie W K, Butt S. Preserving convexity using piecewise cubic interpolation Computer & Graphics, 1991 15(1)15—23
[21] Zhang S. The geometric invariant of implicit curve and surface and geometric continuity between implicit surfaces. Chinese J. Computers, 1999, 22(7): 774—776(in Chinese)(张三元 基于代数曲线段的几何不变量及几何连续性 计算机学报 1999 22(7)774—776)
[22] Fletcher Y, Mcallister F D. Automatic tension adjustment for interpolatory splines IEEE Computer Graphics & Appliment, 1990, 10(1): 10—17
[23] Brodlie W K, Butt S. Preserving convexity using piecewise cubic interpolation Computer & Graphics, 1991 15(1)15—23
[24] Xiuzi Ye, Takashi Maekawa(1999) Differential geometry of intersection curves of two
surfaces Computer Aided Geometric Design 16: 767—788
[25] 石雄辉(2002) 高阶几何连续过渡曲面的造型研究与应用 西北工业大学硕士论文
[26] 梅向明，黄敬之《微分几何》高等教育出版社(第2版) 1988
[27] 梁友栋 曲线、曲面几何连续性问题 数学年刊 1990，11A：3，374-386
[28] Tieru Wu, Yunshi Zhou(2000) On blending of several quadratic algebraic surfaces, Computer Aided Geometric Design 17: 759—766
[29] Makus Schichtel(1993), G~2 blend surfaces and filling of N-sided holes, IEEE Computer Graphics & Appliment, sep. 68—73
[30] Erich Hartmann(1995), Blending an implict with a parametric surface, Computer Aided Geometric Design 12: 825—835
[31] 徐国良(1997)，CAGD中的隐式曲线与曲面，数值计算与计算机应用，199，7(6)：114—124
[32] Hans Hagen, etc.(1996), Surfaces design using triangle patches, Computer Aided Geometric Design 13: 895—904
[33] Guoliang Xu, etc.(2001), C~1 modeling with A-patch from rational trivariate functions, Computer Aided Geometric Design 18: 221—243
[34] Martin R., Shou H., Voiculescu I., Bowyer A., Wang G. (2002) Comparison of interval methods for plotting algebraic curves, CAGD 19: 553—587
[35] Chandler R. E.(1988) A tracking algorithm for implicitly defined curves, IEEE Computer Graphics Appl. 8: 83—89
[36] Duff T. (1992) Interval arithmetic and recursive subdivision for implicit functions and constructive solid geometry, Computer Graphics 26: 131—138
[37] Gopalsamy S., Khandekar D., Mudur S.P.(1991) A new method of valuating compact geometric bounds for use in subdivision algorithms, CAGD 8: 337—356
[38] Hu C. Y., Patrikalakis N. M., Ye X. (1996a) Robust interval solid modeling, Part Ⅰ: Representation, CAD 28: 807—817
[39] Hu C. Y., Patrikalakis N. M., Ye X. (1996b) Robust interval solid modeling, Part Ⅱ: Boundary evaluation, CAD 28: 819—830
[40] Shou H., Martin R., Voiculescu I., Bowyer A., Wang G. (2002) Affine arithmetic in matrix form for algebraic curve drawing, Progress in Natural Science(China)12: 77—80
[41] Snyder J. M. (1992) Interval analysis for computer graphics, Computer Graphics 26: 121—130
[42] Taubin G. (1994) Distance approximation for rasterizing implicit curves, ACM Trans. on
Graphics 13: 3—42
[43] Sederberg T. W., Chen F. (1995) Implicitization using moving curves and surfaces, in: Proc. SIGGRAPH, pp, 301—308
[44] Marco A. & Martinez J.J. (2001) Using polynomial interpolation for implicitizing algebraic curves, CAGD 18: 309—319
[45] Marco A. & Martinez J.J. (2002) Implicitization of rational surfaces by means of polynomial interpolation, CAGD 19: 327—344
[46] Patterson R.R. (2002) Using duality to implieitize and find cusps and inflection point of Bezier curves, CAGD 19: 433—444
[47] Berry T.G.& Patterson R.R. (2001) Implicitization and parametrization of nonsingular cubic surfaces, CAGD 18: 723—738
[48] Chen F. & Sederberg T. (2002) A new implicit representation of a planar curve with high order singularity, CAGD 19: 151—167