Freeth's Nephroid


A strophoid of a circle with the pole O at the center of the circle and the fixed point P on the circumference of the circle. Freeth (1878, pp. 130 and 228) described this and various other strophoids (MacTutor Archive).

It has polar equation


The area enclosed by the outer boundary of the curve is


and the total arc length is


(OEIS A138498), where k=sqrt(2/3), K(k) is a complete elliptic integral of the first kind, E(k) is a complete elliptic integral of the second kind, and Pi(x,k) is a complete elliptic integral of the third kind.

If the line through P parallel to the y-axis cuts the nephroid at A, then angle AOP is 3pi/7, so this curve can be used to construct a regular heptagon.

The curvature and tangential angle are given by


where |_x_| is the floor function.

See also


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Freeth, Rev. T. J. May 8, 1878 communication to the London Math. Soc. referenced as "The Nephroid, Heptagon, &c." Proc. London. Math. Soc. 10, p. 130, 1878. The curve is explicitly described in the Appendix of vol. 10 on p. 228.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 175 and 177-178, 1972.MacTutor History of Mathematics Archive. "Freeth's Nephroid.", N. J. A. Sequences A138498 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Freeth's Nephroid." From MathWorld--A Wolfram Web Resource.

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