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. In a paper published by the London Mathematical Society in 1879, T. J. Freeth described it and various other Strophoids (MacTutor Archive). If the line through $P$ Parallel to the y-Axis cuts the Nephroid at $A$, then Angle $AOP$ is $3\pi/7$, so this curve can be used to construct a regular Heptagon. The Polar equation is

$$r=a[1+2\sin({\textstyle{1\over 2}}\theta)].$$

See also Strophoid

References

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.