Equation of Nephroid

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Theorem

Let $H$ be the nephroid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the outside of a stator $C_2$ of radius $a = 2 b$.

Let $C_2$ be embedded in a cartesian coordinate plane with its center $O$ located at the origin.

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \left({a, 0}\right)$ on the $x$-axis.


Let $\left({x, y}\right)$ be the coordinates of $P$ as it travels over the plane.


The point $P = \left({x, y}\right)$ is described by the parametric equation:

$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$

where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.


Proof

By definition, a nephroid is an epicycloid with $2$ cusps.

Nephroid.png


By Equation of Epicycloid, the equation of $H$ is given by:

$\begin{cases} x & = \left({a + b}\right) \cos \theta - b \cos \left({\left({\dfrac {a + b} b}\right) \theta}\right)\\ y & = \left({a + b}\right) \sin \theta - b \sin \left({\left({\dfrac {a + b} b}\right) \theta}\right) \end{cases}$


When $a = 2 b$ the equation of $H$ is now given by:

$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$

$\blacksquare$


Sources