Freeth's Nephroid

Polar equation:
r = a(1 + 2sin(θ/2))

Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point

If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

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

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

If the line through P parallel to the y-axis cuts the nephroid at A then angle AOP is /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.

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JOC/EFR/BS January 1997

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