Cyclocycloid

The cyclocycloid (in this case an epicycloid) with R = 3, r = 1 and d = 1/2

A 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 (in this case an hypocycloid) 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

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

$y(\theta )=(R+r)\sin \theta -d\sin \left(\frac{R+r}{r}\theta \right).$

where $\theta$ is a parameter (not the polar angle). And r can be positive (represented by a ball rolling outside of a circle) or negative (represented by a ball rolling inside of a circle) 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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