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Parameterize a cycloid: the path of a point on the circumference of a circle of radius a as it rolls without slipping along the positive x-axis.

Parameterize the path of a point at distance b from the center of disk of radius a as it rolls without slipping along the positive x-axis.

Parameterize the path of a point on the circumference of a circle of radius a as it rolls without slipping around another circle of radius a. You may want to experiment with coins to understand the behavior.

 

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