Cycloids

Wikipedia says: "a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping."



If we roll the circle around another circle, we get epicycloids if we run on the outside, and hypocycloids if we run on the inside.

On this page you can play with different sizes of both kinds. Btw, the Spirograph toy is all about cycloids as well as some other so-called roulettes.

The math behind the simple cases demonstrated on this page is quite simple. All you really need for this is to (repeatedly) compute the position of a point on a circle after rotating that circle a certain amount:

If a point P(x, y) on a circle is rotated counter-clockwise by an angle θ around the circle's center located at P(0, 0), then the coordinates of the new point P'(x', y') are given by the following:

• x' = x cos(θ) - y sin(θ)
• y' = x sin(θ) + y cos(θ)

Play with this!!

p.s. When you select hypocycloid with the radius of the inner circle to be one half of that of the outer circle, the cycling point of the inner circle follows a straight line. This is know as a Tusi couple.

Epicycloid Hypocycloid inner circle radius: outer circle radius: Go! Stop!

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