 TOPICS # Strophoid

Let be a curve, let be a fixed point (the pole), and let be a second fixed point. Let and be points on a line through meeting at such that . The locus of and is called the strophoid of with respect to the pole and fixed point . Let be represented parametrically by , and let and . Then the equation of the strophoid is   (1)   (2)

where (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 (4)

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

 curve pole fixed point strophoid line not on line on line oblique strophoid line not on line foot of perpendicular origin to line right strophoid circle center on the circumference Freeth's nephroid

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.

Strophoid

## Cite this as:

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