Strophoid
Let C be a curve, let O be a fixed point (the pole), and let O^' be a second fixed point. Let P and P^' be points on a line through O meeting C at Q such that P^'Q=QP=QO^'. The locus of P and P^' is called the strophoid of C with respect to the pole O and fixed point O^'. Let C be represented parametrically by (f(t),g(t)), and let O=(x_0,y_0) and O^'=(x_1,y_1). Then the equation of the strophoid is
where
| [画像: m=(g-y_0)/(f-x_0). ] |
(3)
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The name strophoid means "belt with a twist," and was proposed by Montucci in 1846 (MacTutor Archive). The polar form for a general strophoid is
If a=pi/2, the curve is a right strophoid. The following table gives the strophoids of some common curves.
See also
Right StrophoidExplore with Wolfram|Alpha
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References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 121, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 51-53 and 205, 1972.Lockwood, E. H. "Strophoids." Ch. 16 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 134-137, 1967.MacTutor History of Mathematics Archive. "Right." https://mathshistory.st-andrews.ac.uk/Curves/Right/.Yates, R. C. "Strophoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 217-220, 1952.Referenced on Wolfram|Alpha
StrophoidCite this as:
Weisstein, Eric W. "Strophoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Strophoid.html