垂足曲线
在曲线微分几何中,踩踏板曲綫是從給定曲綫所創造的曲綫,構造方法像自行車用腳踩踏在原有曲綫上,故稱為踩踏板曲綫,又譯作垂足曲线。给定一个曲线和一个定点P(称为垂足点或踩踏點(Pedal Point))。在曲线的任何一条切线T上,都存在唯一的一个点X,要么是P本身,要么与P形成的直线与T垂直。垂足曲线是符合这种性质的所有点X所组成的集合。
垂足曲线不一定是连通的,例如对于多边形来说,它仅仅是一些孤立的点。
如果P是垂足点,c是曲线的一个参数方程,则垂足曲线的参数方程为:
如果垂足点是原点,则垂足曲线为:
参考文献
- Gray, A. "Pedal Curves." §5.8 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 117-125, 1997.
- Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 46-49 and 204, 1972.
- Lockwood, E. H. "Pedal Curves." Ch. 18 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 152-155, 1967.
- Yates, R. C. "Pedal Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 160-165, 1952.
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