Cyclocycloid

The cyclocycloid with R = 3, r = 1 and d = 1/2

An cyclocycloid is a roulette traced by a point attached to a circle of radius r rolling around, a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.

The red curve is a cyclocycloid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5, r = -3, d = 5).

The parametric equations for a cyclocycloid are

[math]\displaystyle{ x (\theta) = (R + r)\cos\theta - d\cos\left({R + r \over r}\theta\right),\, }[/math]

[math]\displaystyle{ y (\theta) = (R + r)\sin\theta - d\sin\left({R + r \over r}\theta\right).\, }[/math]

where [math]\displaystyle{ \theta }[/math] is a parameter (not the polar angle). And r can be positive or negative depending on whether it is of an Epicycloid or Hypocycloid variety.

The classic Spirograph toy traces out these curves.

See also

• Centered trochoid
• Cycloid
• Epicycloid
• Hypocycloid
• Spirograph

External links

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