TOPICS

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

x=f+/-sqrt(((x_1-f)^2+(y_1-g)^2)/(1+m^2))
(1)
y=g+/-sqrt(((x_1-f)^2+(y_1-g)^2)/(1+m^2)),
(2)

where

 m=(g-y_0)/(f-x_0).
(3)

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

 r=(bsin(a-2theta))/(sin(a-theta)).
(4)

If a=pi/2, the curve is a right strophoid. The following table gives the strophoids of some common curves.

curvepolefixed pointstrophoid
linenot on lineon lineoblique strophoid
linenot on linefoot of perpendicular origin to lineright strophoid
circlecenteron the circumferenceFreeth's nephroid

See also

Right Strophoid

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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." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Right.html.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

Strophoid

Cite this as:

Weisstein, Eric W. "Strophoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Strophoid.html

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