有理二次Bézier形式共轭双曲线段的几何计算
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摘要
考虑有理二次Bézier形式的相互共轭的双曲线的控制顶点之间的关系,给定表示一段双曲线的标准型有理二次Bézier曲线,目标是求出它的共轭双曲线上相应段的控制顶点。首先给出共轭双曲线段的自然定义;接着通过参数变换,将有理二次Bézier形式和一般参数形式进行转换,并把这种转换对应到矩阵,以给出所求控制顶点的显式表达;最后,给出表达式的几何意义,即共轭双曲线段的控制顶点可由原双曲线的控制顶点通过两次线性插值得到。
Consider the relationship between the control points of the conjugate hyperbolas presented as rational quadratic Bézier forms. Given a rational quadratic Bézier curve with standard form presenting a hyperbolic segment, the target is to calculate the control points of the corresponding segment on its conjugate hyperbola. Firstly, the natural definition of the segment of conjugate hyperbola is given. Secondly, parameter conversion is used to transform the hyperbola between the rational quadratic Bézier form and a general parametric form. The transformations correspond to matrices. Thus the explicit expressions of the control points of the conjugate segments are obtained. Finally, the geometric meanings of the expressions are shown. Each control point of the conjugate hyperbola segments can be given by two linear interpolations of the control points of the original hyperbola segment.
Consider the relationship between the control points of the conjugate hyperbolas presented as rational quadratic Bézier forms. Given a rational quadratic Bézier curve with standard form presenting a hyperbolic segment, the target is to calculate the control points of the corresponding segment on its conjugate hyperbola. Firstly, the natural definition of the segment of conjugate hyperbola is given. Secondly, parameter conversion is used to transform the hyperbola between the rational quadratic Bézier form and a general parametric form. The transformations correspond to matrices. Thus the explicit expressions of the control points of the conjugate segments are obtained. Finally, the geometric meanings of the expressions are shown. Each control point of the conjugate hyperbola segments can be given by two linear interpolations of the control points of the original hyperbola segment.
引文
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[13]沈莞蔷,汪国昭.线性双曲拟Bézier曲线的几何图形[J].计算机工程与应用,2014,50(7):10-14,40.
[14]Farin G.Rational quadratic circles are parametrized by chord length[J].Computer Aided Geometric Design,2006,23(9):722-724.
[15]王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育出版社;海德堡:施普林格出版社,2001:35-43.
[16]施法中.反求标准型有理二次Bézier曲线的参数与内权因子[J].计算机辅助设计与图形学学报,1995,7(2):91-95.
[17]Li Yajuan,Wang Guozhao.Two kinds of B-basis of the algebraic hyperbolic space[J].Journal of Zhejiang University Science A,2005,6(7):750-759.
[18]Xu Gang,Wang Guzhao.AHT Bézier curves and NUAHT B-spline curves[J].Journal of Computer Science and Technology,2007,22(4):597-607.
[19]沈莞蔷,汪国昭,徐红林.线性p-Bézier曲线的几何形状[J].计算机辅助设计与图形学学报,2014,26(8):1211-1218.
[2]赵文彬,谢晓薇,杨福增,等.基于二次有理Bézier曲线拟合的颅骨识别技术研究[J].生物医学工程学杂志,2008,25(2):280-284.
[3]Hu Qianqian.Approximating conic sections by constrained Bézier curves of arbitrary degree[J].Journal of Computational and Applied Mathematics,2012,236(11):2813-2821.
[4]Walton D J,Meek D S.G2 curve design with planar quadratic rational Bézier spiral segments[J].International Journal of Computer Mathematics,2013,90(2):325-340.
[5]陈军,王国瑾.规避障碍物的G2连续有理二次Bézier样条曲线[J].计算机辅助设计与图形学学报,2011,23(4):582-585.
[6]闫宏业,刘莉,廖志刚,等.采用改进的修正双曲线法预测高速铁路路基沉降[J].铁道建筑,2011,12:92-94.
[7]孙敬松,周广胜.基于叶面积指数改进的直角双曲线模型在玉米农田生态系统中的应用[J].生态学报,2013,33(7):2182-2188.
[8]施剑,吴志强,崔三元,等.南黄海盆地地震勘探中的非双曲线时差分析[J].石油地球物理勘探,2013,48(2):192-199.
[9]Hu Qianqian,Wang Guojin.Geometric meanings of the parameters on rational conic segments[J].Science in China Series A:Mathematics,2005,48(9):1209-1222.
[10]Isabelle C H,Gudrun A,Victoriah M.Optimal parameterization of rational quadratic curves[J].Computer Aided Geometric Design,2009,26(7):725-732.
[11]Lee E T Y.The rational Bézier representation for conics[C]//Farin G E.Geometric Modeling:Algorithms and New Trends,vol 14.Philadelphia:Society for Industrial and Applied Mathematics Publications,1987:3-19.
[12]Xu Chengdong,Kim T W,Farin G.The eccentricity of conic sections formulated as rational Bézier quadratics[J].Computer Aided Geometric Design,2010,27(6):458-460.
[13]沈莞蔷,汪国昭.线性双曲拟Bézier曲线的几何图形[J].计算机工程与应用,2014,50(7):10-14,40.
[14]Farin G.Rational quadratic circles are parametrized by chord length[J].Computer Aided Geometric Design,2006,23(9):722-724.
[15]王国瑾,汪国昭,郑建民.计算机辅助几何设计[M].北京:高等教育出版社;海德堡:施普林格出版社,2001:35-43.
[16]施法中.反求标准型有理二次Bézier曲线的参数与内权因子[J].计算机辅助设计与图形学学报,1995,7(2):91-95.
[17]Li Yajuan,Wang Guozhao.Two kinds of B-basis of the algebraic hyperbolic space[J].Journal of Zhejiang University Science A,2005,6(7):750-759.
[18]Xu Gang,Wang Guzhao.AHT Bézier curves and NUAHT B-spline curves[J].Journal of Computer Science and Technology,2007,22(4):597-607.
[19]沈莞蔷,汪国昭,徐红林.线性p-Bézier曲线的几何形状[J].计算机辅助设计与图形学学报,2014,26(8):1211-1218.
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