高阶几何连续过渡曲面的造型研究与应用
|
摘要
过渡曲面在CAD／CAM技术中具有十分重要的地位，研究高阶几何连续过渡曲面的构造理论与方法是当前的研究热点之一。本论文主要研究曲面间具有三阶及更高阶几何连续过渡曲面的构造问题，主要结果如下：
首先对曲面造型研究的现状、主流和趋势等作了总的概述。在此基础上根据几何连续与过渡曲面的密切相关性，重点介绍过渡曲面模型与几何连续的理论与性质。
其次，较系统的介绍和分析了过渡曲面构造的各种方法。指出了一般过渡曲面算法中存在的主要问题。
第三，依据几何连续的“重新标度连续”定义，结合几何过渡方法，成功地建立了隐式曲面与参数曲面间几何连续过渡构造方法。得到隐式曲面与参数曲面间G~1、G~2、G~3过渡曲面。
第四，从几何不变量与几何连续性的关系入手，导出并证明了曲面之间三阶及高阶几何连续的充要条件。将其充分性与超限插值方法相结合，成功地构造出两个参数曲面间的高阶几何连续过渡曲面。
最后，为满足曲面造型系统集成的需要，在保证曲率、曲率导数和挠率精度情况下，将上述研究结果应用于曲面放样逼近方法，得到参数过渡曲面转换成B样条曲面或NURBS曲面的形式。
Blending surface plays an important role in CAD/CAM, and the problem of blending surface with high-degree geometric continuity is a key question. The paper concerns how to construct a blending surface between two surfaces.
(a) It summarizes geometric modeling on its development, its new methods, and further trend, and gives the theory of geometric continuity and blending surface.
(b) It systematically introduces main blending method, and points out the defect of blending algorithm in general.
(c) G1, G2 and G3 blending surfaces have successfully been constructed by combining rescaling continuity with geometric blending methods.
(d) From the relation of geometric invariant with blending surface, a sufficient and necessary condition is derived. Then a high-degree geometric continuity blending surface is obtained with transfinite interpolate method.
(e) G3 blending surface with approximation can be constructed by lofting if geometric invariant errors are given, the surface can be converted to B-spline, NURBS form for the purpose of system aggregation.
首先对曲面造型研究的现状、主流和趋势等作了总的概述。在此基础上根据几何连续与过渡曲面的密切相关性，重点介绍过渡曲面模型与几何连续的理论与性质。
其次，较系统的介绍和分析了过渡曲面构造的各种方法。指出了一般过渡曲面算法中存在的主要问题。
第三，依据几何连续的“重新标度连续”定义，结合几何过渡方法，成功地建立了隐式曲面与参数曲面间几何连续过渡构造方法。得到隐式曲面与参数曲面间G~1、G~2、G~3过渡曲面。
第四，从几何不变量与几何连续性的关系入手，导出并证明了曲面之间三阶及高阶几何连续的充要条件。将其充分性与超限插值方法相结合，成功地构造出两个参数曲面间的高阶几何连续过渡曲面。
最后，为满足曲面造型系统集成的需要，在保证曲率、曲率导数和挠率精度情况下，将上述研究结果应用于曲面放样逼近方法，得到参数过渡曲面转换成B样条曲面或NURBS曲面的形式。
Blending surface plays an important role in CAD/CAM, and the problem of blending surface with high-degree geometric continuity is a key question. The paper concerns how to construct a blending surface between two surfaces.
(a) It summarizes geometric modeling on its development, its new methods, and further trend, and gives the theory of geometric continuity and blending surface.
(b) It systematically introduces main blending method, and points out the defect of blending algorithm in general.
(c) G1, G2 and G3 blending surfaces have successfully been constructed by combining rescaling continuity with geometric blending methods.
(d) From the relation of geometric invariant with blending surface, a sufficient and necessary condition is derived. Then a high-degree geometric continuity blending surface is obtained with transfinite interpolate method.
(e) G3 blending surface with approximation can be constructed by lofting if geometric invariant errors are given, the surface can be converted to B-spline, NURBS form for the purpose of system aggregation.
引文
1. S. Bedi & G.W. Vickers (1989) Surface lofting and smoothing with skeletal-lines. CAGD 6, 87-96.
2. L. Holmstrom(1987) Piecewise quadric blending of implicitly defined surfaces.CAGD 4,171-189.
3. T. Hertmann & G. Lukacs et, al(1999) Geometrical criteria on the higher order smoothness of composite surfaces. CAGD 16,907-911.
4. Li Jinggong et, al(1990) G~(n-1)-functional splines for interpolation and approximation of curves, surfaces and solids. CAGD 7, 209-220.
5. M. A. Sanglikar et, al(1990) Modeling rolling ball blends for computer aided geometric design. CAGD 7,399-414.
6. T. Garrty, et, al(1991) Geometric continuity. CAGD 8, (1991). 51-65
7. Hu Shi-Min(1996) Conversion of a triangular Bezier patch into three rectangular Bezier patch. CAGD 13, 219-226.
8. M-L Mazure(1994) Geometric contact for curves and surfaces. CAGD 11,177-195.
9. F. Arbab(1990) Set Model and Boolean Operations for Solids and Assemblies.IEEE Computer graphics & Applications. November 1990, 76-86.
10. M. G Bloor & M. J Wilson Generating blend surfaces using partial differential equations. CAD 1989, 21,165-171.
11. L.A. Piegl & W. Tiller Computing offsets of NURBS curves and surfaces CAD 1999,31,147-156.
12. Markus Schichtel G~2 Blend surfaces and filling of N-sided holes. IEEE Computer graphics & Applications. September, 1993, 68-73.
13. Erich Hartmann Blending an implicit with a parametric surface. CAGD 1995,12,825-835
14. J. Vida, R. R. Martin, T. Varady A survey of blending methods that use parametric surfaces. CAD vol.26 No.5 341-365.
15. C. Hoffmann and J. Hopcroft Quadratic blending surfaces. CAD 1986 vol. No.6
301-306
16. M.G Bloor & M. J Wilson, Using partial differential equations to generate free-from surfaces. CAD 1990, vol. 22, No.4 202-212.
17. B.K. Choi&S.Y. Ju Constant-radius blending in surface modeling. CAD 1989, vol.21, No.4 213-220.
18. R.E. Barnhill, G. E. Farin, Qian Chen Constant-radius blending of parametric surfaces. Computing Suppl. 1993, 8,1-20.
19. J. Pegna, F-E. Wolter Geometrical criteria to guarantee curvature continuity of blend surfaces. ASME Journal of Mechanical Design 1992, vol. 114, 201-210.
20. T. Maekawa An overview of offset curves and surfaces. CAD 1999, 31,165-173.
21. N.M. Patrikalakis Approximate conversion of rational splines. CAGD 1989,6,155-165.
22. L. Bardis, N.M. Patrikalakis Approximate conversion of rational B-spline patches. CAGD 1989, 6,189-204.
23. E. Hartmann Numerical parameterization of curves and surfaces. CAGD 2000,17,251-266.
24. R. T. Farouki, R. Sverrisson Approximation of rolling-ball blends for free-form parametric surfaces. CAD vol. 28, No. 11, 871-878.
25. E. Hartmann Blending of implicit surfaces with functional splines. CAD 1990,vol. 22, No.8, 500-506.
26. F-E. Wolter and S.T. Touhy Approximation of high-degree and procedural curves. Engineering with computers 1992, 8, 61-80.
27. L.A. Piegl and W. Tiller Cross-sectional design with boundary constraints.Engineering with computers 1999,15,171-180.
28. Y. Tokuyama Skinning-surface generation based on spine-curve control. The visual computer 2000,16,134-140.
29. Gregory & M. Nielson Scattered data modeling. IEEE Computer graphics & Applications. January 1993, 61-70.
30. S.T. Touhy and L. Bardis Low-degree approximation of high-degree B-spline surface. Engineering with computers 1993,9,198-209.
31. J.Mitchell Topological obstructions to blending algorithms. CAGD
2000,17,673-694.
32. Marta Szilvasi-Nagy et. al Genereting curves and swept surfaces by blended circles. CAGD 2000,17,197-206.
33. L.Bardis, N.M. Patrikalakis Surface approximation with rational B-splines.Engineering with computers 1990, 6, 223-235.
34. P.C. Filkins, S.T. Touhy, N.M. Patrikalakis Computational methods for blending surface approximation. Engineering with computers 1993,9,49-62.
35. G.. Farin Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. Academic Press, Second Edition 1988.
36．梁友栋 曲线、曲面几何连续性问题。数学年刊 1990，11A：3，374-386。
37．康宝生等 任意曲面间跟踪求交的有效算法。计算机辅助几何设计与图形学，1998年第六期。
38．孟雅琴、叶正麟、彭国华 平面曲线等距线逼近的仿射变换方法。西北工业大学学报 1998年第四期。
39．叶正麟、孟雅琴、刘克轩 可展Bezier曲面。数值计算与计算机应用。1997年第二期
40．梅向明、黄敬之编《微分几何》高等教育出版社，1988年2月第二版。
41．多卡模 著，田畴等译，胡和生、姜国英校《曲线和曲面的微分几何学》。上海科学技术出版社，1988年10月第一版。
42．朱心雄等 著《自由曲线曲面造型技术》。科学出版社，2000年1月第一版。
43．施法中 著《计算机辅助几何设计与非均匀有理B样条》。北京航空航天大学出版社，1994。
44．张文贵译，朱鼎勋校[前苏联]《微分几何习题集》。北京师范大学出版社，1983年第一版。
45．张永曙、刘克轩、蒋大为等著《计算机辅助几何设计的数学方法》。西北工业大学出版社，1986。
46．胡冠章 著《应用近世代数》。清华大学出版社，1999年2月第二版。
47．孙家昶著《样条函数与计算几何》科学出版社，1982。
48．王国瑾、汪国昭、郑建民著《计算机辅助几何设计》。高等教育出版社、施普林格出版社，2001。
49．王省富、彭国华等编著《样条函数及其应用》。西北工业大学出版社1989年9月第一版。
2. L. Holmstrom(1987) Piecewise quadric blending of implicitly defined surfaces.CAGD 4,171-189.
3. T. Hertmann & G. Lukacs et, al(1999) Geometrical criteria on the higher order smoothness of composite surfaces. CAGD 16,907-911.
4. Li Jinggong et, al(1990) G~(n-1)-functional splines for interpolation and approximation of curves, surfaces and solids. CAGD 7, 209-220.
5. M. A. Sanglikar et, al(1990) Modeling rolling ball blends for computer aided geometric design. CAGD 7,399-414.
6. T. Garrty, et, al(1991) Geometric continuity. CAGD 8, (1991). 51-65
7. Hu Shi-Min(1996) Conversion of a triangular Bezier patch into three rectangular Bezier patch. CAGD 13, 219-226.
8. M-L Mazure(1994) Geometric contact for curves and surfaces. CAGD 11,177-195.
9. F. Arbab(1990) Set Model and Boolean Operations for Solids and Assemblies.IEEE Computer graphics & Applications. November 1990, 76-86.
10. M. G Bloor & M. J Wilson Generating blend surfaces using partial differential equations. CAD 1989, 21,165-171.
11. L.A. Piegl & W. Tiller Computing offsets of NURBS curves and surfaces CAD 1999,31,147-156.
12. Markus Schichtel G~2 Blend surfaces and filling of N-sided holes. IEEE Computer graphics & Applications. September, 1993, 68-73.
13. Erich Hartmann Blending an implicit with a parametric surface. CAGD 1995,12,825-835
14. J. Vida, R. R. Martin, T. Varady A survey of blending methods that use parametric surfaces. CAD vol.26 No.5 341-365.
15. C. Hoffmann and J. Hopcroft Quadratic blending surfaces. CAD 1986 vol. No.6
301-306
16. M.G Bloor & M. J Wilson, Using partial differential equations to generate free-from surfaces. CAD 1990, vol. 22, No.4 202-212.
17. B.K. Choi&S.Y. Ju Constant-radius blending in surface modeling. CAD 1989, vol.21, No.4 213-220.
18. R.E. Barnhill, G. E. Farin, Qian Chen Constant-radius blending of parametric surfaces. Computing Suppl. 1993, 8,1-20.
19. J. Pegna, F-E. Wolter Geometrical criteria to guarantee curvature continuity of blend surfaces. ASME Journal of Mechanical Design 1992, vol. 114, 201-210.
20. T. Maekawa An overview of offset curves and surfaces. CAD 1999, 31,165-173.
21. N.M. Patrikalakis Approximate conversion of rational splines. CAGD 1989,6,155-165.
22. L. Bardis, N.M. Patrikalakis Approximate conversion of rational B-spline patches. CAGD 1989, 6,189-204.
23. E. Hartmann Numerical parameterization of curves and surfaces. CAGD 2000,17,251-266.
24. R. T. Farouki, R. Sverrisson Approximation of rolling-ball blends for free-form parametric surfaces. CAD vol. 28, No. 11, 871-878.
25. E. Hartmann Blending of implicit surfaces with functional splines. CAD 1990,vol. 22, No.8, 500-506.
26. F-E. Wolter and S.T. Touhy Approximation of high-degree and procedural curves. Engineering with computers 1992, 8, 61-80.
27. L.A. Piegl and W. Tiller Cross-sectional design with boundary constraints.Engineering with computers 1999,15,171-180.
28. Y. Tokuyama Skinning-surface generation based on spine-curve control. The visual computer 2000,16,134-140.
29. Gregory & M. Nielson Scattered data modeling. IEEE Computer graphics & Applications. January 1993, 61-70.
30. S.T. Touhy and L. Bardis Low-degree approximation of high-degree B-spline surface. Engineering with computers 1993,9,198-209.
31. J.Mitchell Topological obstructions to blending algorithms. CAGD
2000,17,673-694.
32. Marta Szilvasi-Nagy et. al Genereting curves and swept surfaces by blended circles. CAGD 2000,17,197-206.
33. L.Bardis, N.M. Patrikalakis Surface approximation with rational B-splines.Engineering with computers 1990, 6, 223-235.
34. P.C. Filkins, S.T. Touhy, N.M. Patrikalakis Computational methods for blending surface approximation. Engineering with computers 1993,9,49-62.
35. G.. Farin Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. Academic Press, Second Edition 1988.
36．梁友栋 曲线、曲面几何连续性问题。数学年刊 1990，11A：3，374-386。
37．康宝生等 任意曲面间跟踪求交的有效算法。计算机辅助几何设计与图形学，1998年第六期。
38．孟雅琴、叶正麟、彭国华 平面曲线等距线逼近的仿射变换方法。西北工业大学学报 1998年第四期。
39．叶正麟、孟雅琴、刘克轩 可展Bezier曲面。数值计算与计算机应用。1997年第二期
40．梅向明、黄敬之编《微分几何》高等教育出版社，1988年2月第二版。
41．多卡模 著，田畴等译，胡和生、姜国英校《曲线和曲面的微分几何学》。上海科学技术出版社，1988年10月第一版。
42．朱心雄等 著《自由曲线曲面造型技术》。科学出版社，2000年1月第一版。
43．施法中 著《计算机辅助几何设计与非均匀有理B样条》。北京航空航天大学出版社，1994。
44．张文贵译，朱鼎勋校[前苏联]《微分几何习题集》。北京师范大学出版社，1983年第一版。
45．张永曙、刘克轩、蒋大为等著《计算机辅助几何设计的数学方法》。西北工业大学出版社，1986。
46．胡冠章 著《应用近世代数》。清华大学出版社，1999年2月第二版。
47．孙家昶著《样条函数与计算几何》科学出版社，1982。
48．王国瑾、汪国昭、郑建民著《计算机辅助几何设计》。高等教育出版社、施普林格出版社，2001。
49．王省富、彭国华等编著《样条函数及其应用》。西北工业大学出版社1989年9月第一版。