Equation of Deltoid

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Theorem

Let $H$ be the deltoid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the inside of a stator $C_2$ of radius $a = 3 b$.

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:

$\begin{cases} x & = 2 b \cos \theta + b \cos 2 \theta \\ y & = 2 b \sin \theta - b \sin 2 \theta \end{cases}$

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 deltoid is a hypocycloid with $3$ cusps.

Deltoid.png


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

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

From Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii, this can be generated by a rotor $C_1$ of radius $\dfrac 1 3$ the radius of the stator.

Thus $a = 3 b$ and the equation of $H$ is now given by:

$\begin{cases} x & = 2 b \cos \theta + b \cos 2 \theta \\ y & = 2 b \sin \theta - b \sin 2 \theta \end{cases}$

$\blacksquare$


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