23 April 2018


The hypotrochoid is a transcendental plane curve, corresponding to a fixed point on a moving circle that rolls without slipping over and internally around another circle, called the director circle. This curve can be drawn through the spirograph.

The parametric equations for a hypotrochoid are:

\(x=\left( R-r \right) \cos t+d \cos \left( \frac{R-r}{r}t \right)\)

\(y=\left( R-r \right) \sin t-d \sin \left( \frac{R-r}{r}t \right)\)

where \(R\) is the radius of the director circle (fixed), \(r\) is the radius of the mobile circle, \(d\) is the distance from the center point of the mobile circle and \(t\) is the parameter of the angle.

The hypocycloid (\(d=r\)) and the ellipse (\(R=2r\)) are particular cases of the hypotrochoid.

In the applet below, click in the Start button to begin the animation that shows the drawing of an hypotrochoid. You can use the Stop button to stop the animation at any point. Click in the Reset  button to erase the last curve. You can change the values of R, r and d in the sliders in order to obtain different kinds of hypotrochoids. You can also hide the circles and the line segment in order to see only the curve.

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