Hypocycloid derivation of parametric equations and examples of hypocycloid animations.

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Hypocycloid derivation of parametric equations and examples of hypocycloid animations.

We begin with an animation of a hypocycloid curve generated by a smaller circle rolling within a larger circle. The animation of the smaller circle rolling inside the larger circle produces a three pointed hypocycloid, because the ratio of the circumferences of the larger circle to the smaller circle is 3:1.

We freeze the hypocycloid animation and begin to label the diagram in order to derive the parametric equations for the hypocycloid generated by rotations of the smaller circle within the larger circle.

We use trigonometry to derive the parametric equations for the hypocycloid in general, where the radius of the larger circle is R and the radius of the smaller circle is r.

Next, we apply our parametric equations to hypocycloid examples. The first is the same as the introductory animation: a three-pointed hypocycloid generated from R=3 and r=1. After finding the parametric equations for the hypocycloid, we show an animation of the curve.

Finally, we apply the hypocycloid parametric equations to the case for which R=3 and r=1.1. In this case, the curve is not closed after a 2pi rotation of the parameter t. We compute how many rotations before the parametric curve is complete, then we finish with an animation of the hypocycloid, which takes 11 complete rotations of the parameter t to complete.