# Nephroid

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 52A10 [MSN][ZBL]

An epicycloid with parameter $m=2$; an algebraic plane curve with equation $$ x= 3r \cos\theta-r\cos\left[3\theta\right] \,, $$ $$ y= 3r \sin\theta-r\sin\left[3\theta\right] \ . $$

The nephroid is the catacaustic of the cardioid with respect to a cusp, and of a circle with respect to a point at infinity; the evolute of a nephroid is another nephroid.

The **nephroid of Freeth** is the strophoid of a circle with respect to its centre and a point on the circumference. It has equation
$$
r = a(1 + 2\sin(\theta/2)) \ .
$$

#### References

- J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002

**How to Cite This Entry:**

Nephroid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Nephroid&oldid=51379