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
| 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
(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
![画像:(sqrt(2)[9-3cost+9sin(1/2t)])/([7-3cost+8sin(1/2t)]^(3/2))](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FreethsNephroid/Inline19.svg){kind=link}
![画像:-1/4pi+t+tan^(-1)[sec(1/2t)+2tan(1/2t)]+pi|_(pi+t)/(2pi)_|,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FreethsNephroid/Inline22.svg){kind=link}
where |_x_| is the floor function.
See also
StrophoidExplore with Wolfram|Alpha
More things to try:
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." https://mathshistory.st-andrews.ac.uk/Curves/Freeths/.Sloane, N. J. A. Sequences A138498 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Freeth's NephroidCite this as:
Weisstein, Eric W. "Freeth's Nephroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FreethsNephroid.html