可展同坡曲面及其在边坡建模中的应用
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摘要
三维建模是数字化边坡的重要环节.作为一种表达边坡的曲面模型——同坡曲面,其数学表达和几何特性是用其构建三维边坡的基础.首先,用包络线概念推导并化简得到一个易于计算的同坡曲面参数方程(仅需计算导线的一阶导的二次多项式).其次,证明了同坡曲面是可展的直纹曲面,并以此为基础,分析了此类曲面的坡度、连续性和导线的切矢与母线的夹角等几何特性.最后,将分析结果运用到参数化网格坡面片、分段边坡的三维建模中,并给出了工程适用条件.通过一段弯道路基工程实例,对所提出的建模方法进行了验证,结果表明:此方法便于数字化实现、造型准确美观,可以满足路基边坡的三维建模需要.
3 D modeling plays key role in digital slope design. As one of 3 D representation of slope surface,called iso-slope surface, the study of its mathematical representation and geometric characteristics is the basis for building 3 D slope with it. In this paper, we firstly use the concept of envelope to derive a parametric equation of the iso-slope surface and make it easy to use by simplifying the equation, in which only the quadratic polynomial of first-order derivative of the directrix curve needs to be calculated. Secondly, we prove that the iso-slope surface is a developable ruled surface. Based on this, the geometrical characteristics of such surface are analyzed, including slope degree, continuity and angle between the directrix tangent vector and the rule line. Finally, these characteristics are applied to the 3 D modeling of parametric mesh slope surface patch, segmentation slope. Also, the engineering conditions are given. By a case study of curved subgrade, the modeling method proposed is verified. The results show that this method is convenient to be realized digitally, and the output model is accurate and elegant. Furthermore, it can meet the needs of3 D slope modeling.
3 D modeling plays key role in digital slope design. As one of 3 D representation of slope surface,called iso-slope surface, the study of its mathematical representation and geometric characteristics is the basis for building 3 D slope with it. In this paper, we firstly use the concept of envelope to derive a parametric equation of the iso-slope surface and make it easy to use by simplifying the equation, in which only the quadratic polynomial of first-order derivative of the directrix curve needs to be calculated. Secondly, we prove that the iso-slope surface is a developable ruled surface. Based on this, the geometrical characteristics of such surface are analyzed, including slope degree, continuity and angle between the directrix tangent vector and the rule line. Finally, these characteristics are applied to the 3 D modeling of parametric mesh slope surface patch, segmentation slope. Also, the engineering conditions are given. By a case study of curved subgrade, the modeling method proposed is verified. The results show that this method is convenient to be realized digitally, and the output model is accurate and elegant. Furthermore, it can meet the needs of3 D slope modeling.
引文
[1]蒲小琼,陈玲,尹湘云.画法几何及水利土建制图[M].武汉:武汉大学出版社,2015:186-195.Pu Xiaoqiong,Chen Ling,Yin Xiangyun.Descriptive Geometry and Hydraulic&Civil Engineering Drawing[M].Wuhan:Wuhan University Press,2015:186-195.
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[4]李军成,宋来忠.工程中同坡曲面的计算机辅助几何设计[J].武汉大学学报(工学版),2010,43(1):98-101.Li Juncheng,Song Laizhong.Computer aided geometric design of identical slope gradient surface in engineering[J].Engineering Journal of Wuhan University,2010,43(1):98-101.
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[8]Malecek K,Szarka J,Szarkova D.Surfaces with constant slope and their generalization[J].The Journal of Polish Society for Geometry Engineering Graphics,2009,19(2):67-77.
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[12]叶其孝,沈永欢.实用数学手册[M].第2版.北京:科学出版社,2006:402-403.Ye Qixiao,Shen Yonghuan.Handbook of Practical Mathematics[M].Second Edition.Beijing:Science Press,2006:402-403.
[13]Manfredo do Carmo.Differential Geometry of Curves and Surfaces[M].Prentice Hall,1976.
[14]张金水,张廷楷.道路勘测与设计[M].第3版.上海:同济大学出版社,2015:128-149.Zhang Jinshui,Zhang Tingkai.Road Survey and Design[M].Third Edition.Shanghai:Tongji University Press,2015:128-149.
[2]陈永喜.同坡曲面的数学模型及计算机辅助几何设计[J].工程图学学报,1998,(1):48-54.Chen Yongxi.Mathematical model and computer aided geometric design of identical slope gradient surfaces[J].Journal of Enginering Graphics,1998,(1):48-54.
[3]刘青科,毛君.同坡曲面的方程及其在弯斜路面边坡设计和绘图中的应用[J].武汉大学学报(工学版),2006,39(4):111-114.Liu Qingke,Mao Jun.Equation of same slope curved surface and its application to design and drawing of bend and inclined road surface slope[J].Engineering Journal of Wuhan University,2006,39(4):111-114.
[4]李军成,宋来忠.工程中同坡曲面的计算机辅助几何设计[J].武汉大学学报(工学版),2010,43(1):98-101.Li Juncheng,Song Laizhong.Computer aided geometric design of identical slope gradient surface in engineering[J].Engineering Journal of Wuhan University,2010,43(1):98-101.
[5]Munteanu M I,Nistor A I.A new approach on constant angle surfaces in E3[J].Turkish Journal of Mathematics,2008,33(2):169-178.
[6]Nistor A I.Certain constant angle surfaces constructed on curves[J].International Electronic Journal of Geometry,2011,1(4):79-87.
[7]Wang Xiaoliu,Chao Xiaoli.Constant angle surfaces constructed on curves[J].Journal of Southeast University,2013,29(4):470-472.
[8]Malecek K,Szarka J,Szarkova D.Surfaces with constant slope and their generalization[J].The Journal of Polish Society for Geometry Engineering Graphics,2009,19(2):67-77.
[9]Pottmann H,Eigensatz M,Vaxman A,et al.Architectural geometry[J].Computers&Graphics,2015,47(6):145-164.
[10]Decaudin P,Dan J,Wither J,et al.Virtual garments:A fully geometric approach for clothing design[J].Computer Graphics Forum,2006,25(3):625-634.
[11]Peternell M,Steiner T.A geometric idea to solve the eikonal equation[C]//Spring Conference on Computer Graphics SCCG,2005:43-48.
[12]叶其孝,沈永欢.实用数学手册[M].第2版.北京:科学出版社,2006:402-403.Ye Qixiao,Shen Yonghuan.Handbook of Practical Mathematics[M].Second Edition.Beijing:Science Press,2006:402-403.
[13]Manfredo do Carmo.Differential Geometry of Curves and Surfaces[M].Prentice Hall,1976.
[14]张金水,张廷楷.道路勘测与设计[M].第3版.上海:同济大学出版社,2015:128-149.Zhang Jinshui,Zhang Tingkai.Road Survey and Design[M].Third Edition.Shanghai:Tongji University Press,2015:128-149.