Geometric Design Using Space Curves with Rational Rotation-Minimizing Frames
Geometric Design Using Space Curves with Rational Rotation-Minimizing Frames
复制标题
使用空间曲线和合理旋转最小化框架的几何设计
DOI：
10.1007/978-3-642-11620-9_13
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
A. Sestini
中科院分区：
文献类型：
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作者：
R. Farouki;Carlotta Giannelli;A. Sestini
给定空间曲线（t）上的旋转最小化自适应框架（f1（t），f2（t），f3（t））的特征在于框架向量f1与切线t =r′/|r′|，而帧角速度ω保持沿其沿着的零分量，即，ω·t 0。这种框架在构造扫掠曲面和确定刚体沿给定空间路径沿着运动的方向时非常有用。最近，具有旋转最小化框架的五次多项式曲线（五次RRMF曲线）的存在性已被证明。这些RRMF曲线必然是Pythagorean-hodograph（PH）空间曲线，满足空间PH曲线的Hopf映射表示的复系数之间的某些非线性约束。本文给出了用G1空间Hermite数据插值设计五次RRMF曲线的初步结果。这个问题涉及到解决一个非线性方程组在六个复杂的未知数。该问题的求解是通过一个半数值格式来实现的，该格式将问题归结为计算某个一元多项式的正真实的根。五次RRMFG 1Hermite插值具有一个剩余角自由度，它对曲线形状有很大影响.计算实例来说明该方法和所得的五次RRMF曲线。
A rotation–minimizing adapted frame (f1(t),f2(t),f3(t)) on a given space curver(t) is characterized by the fact that the frame vectorf1coincides with the tangentt=r′/ |r′|, while the frame angular velocityωmaintains a zero component along it, i.e.,ω·t≡ 0. Such frames are useful in constructing swept surfaces and specifying the orientation of a rigid body moving along a given spatial path. Recently, the existence of quintic polynomial curves that haverationalrotation–minimizing frames (quintic RRMF curves) has been demonstrated. These RRMF curves are necessarily Pythagorean–hodograph (PH) space curves, satisfying certain non–linear constraints among the complex coefficients of the Hopf map representation for spatial PH curves. Preliminary results on the design of quintic RRMF curves by the interpolation ofG1spatial Hermite data are presented in this paper. This problem involves solving a non–linear system of equations in six complex unknowns. The solution is obtained by a semi–numerical scheme, in which the problem is reduced to computing positive real roots of a certain univariate polynomial. The quintic RRMFG1Hermite interpolants possess one residual angular degree of freedom, which can strongly influence the curve shape. Computed examples are included to illustrate the method and the resulting quintic RRMF curves.
DOI：
10.1007/978-3-540-73843-5
发表时间：
2007
期刊：
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影响因子：
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作者：
Ralph Robert Martin;M. Sabin;J. Winkler
通讯作者：
Ralph Robert Martin;M. Sabin;J. Winkler