Freeth's Nephroid

Polar equation:
r=a(1+2sin(θ/2))r = a(1 + 2\sin( \theta /2))


This is a strophoid of a circle with the pole OO at the centre of the circle and the fixed point PP on the circumference of the circle.

In the picture above, OO is the origin and PP is the node where the curve crosses itself three times.

If the line through PP parallel to the yy-axis cuts the nephroid at AA then angle AOPAOP is 3π /7 . This can be used to construct a regular 7 sided figure.

T J Freeth (1819-1904) was an English mathematician. In a paper published by the London Mathematical Society in 1879 he described various strophoids, including the strophoid of a trisectrix.