Trochoid

General description

A more general approach would define a trochoid as the locus of a point ${\displaystyle (x,y)}$ orbiting at a constant rate around an axis located at ${\displaystyle (x^{\prime },y^{\prime })}$,

${\displaystyle x=x^{\prime }+r_1\cos ({\omega }_{1}t+{\varphi }_1),y=y^{\prime }+r_1\sin ({\omega }_{1}t+{\varphi }_1),r_1>0,}$

which axis is being translated in the x-y-plane at a constant rate in either a straight line,

${\displaystyle \begin{array}{l}{x}^{\prime }=x_0+v_{2x}t,y^{\prime }=y_0+v_{2y}t\\ \therefore x=x_0+r_1\cos ({\omega }_{1}t+{\varphi }_1)+v_{2x}t,y=y_0+r_1\sin ({\omega }_{1}t+{\varphi }_1)+v_{2y}t,\end{array}}$

or a circular path (another orbit) around ${\displaystyle (x_0,y_0)}$ (the hypotrochoid/epitrochoid case),

${\displaystyle \begin{array}{l}{x}^{\prime }=x_0+r_2\cos ({\omega }_{2}t+{\varphi }_2),y^{\prime }=y_0+r_2\sin ({\omega }_{2}t+{\varphi }_2),r_2\ge 0\\ \therefore x=x_0+r_1\cos ({\omega }_{1}t+{\varphi }_1)+r_2\cos ({\omega }_{2}t+{\varphi }_2),y=y_0+r_1\sin ({\omega }_{1}t+{\varphi }_1)+r_2\sin ({\omega }_{2}t+{\varphi }_2),\end{array}}$

The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, ${\displaystyle {\omega }_1/{\omega }_2}$, is a rational number, say ${\displaystyle p/q}$, where ${\displaystyle p}$ & ${\displaystyle q}$ are coprime, in which case, one period consists of ${\displaystyle p}$ orbits around the moving axis and ${\displaystyle q}$ orbits of the moving axis around the point ${\displaystyle (x_0,y_0)}$. The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius ${\displaystyle r_1}$ while it is rolled on the perimeter of a stationary circle of radius ${\displaystyle R}$, have the following properties:

${\displaystyle \begin{array}{lcl}\text{epicycloid: }& {\omega }_1/{\omega }_2& =p/q=r_2/r_1=R/r_1+1,|p-q|\text{ cusps}\\ \text{hypocycloid: }& {\omega }_1/{\omega }_2& =p/q=-r_2/r_1=-(R/r_1-1),|p-q|=|p|+|q|\text{ cusps}\end{array}}$

where ${\displaystyle r_2}$ is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".