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Freeth's Nephroid

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FreethsNephroid

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

r=a[1+2sin(1/2theta)].
(1)

The area enclosed by the outer boundary of the curve is

A=a^2(8+3pi),
(2)

and the total arc length is

s=8/3sqrt(3)a[3E(k)-3K(k)+4Pi(-1/3,k)]
(3)
=21.203405...a
(4)

(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

kappa(t)=(sqrt(2)[9-3cost+9sin(1/2t)])/([7-3cost+8sin(1/2t)]^(3/2))
(5)
phi(t)=-1/4pi+t+tan^(-1)[sec(1/2t)+2tan(1/2t)]+pi|_(pi+t)/(2pi)_|,
(6)

where |_x_| is the floor function.

See also

Strophoid

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References

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." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Freeths.html.Sloane, 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. https://mathworld.wolfram.com/FreethsNephroid.html

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