The Cycloid

Imagine a point on a wheel (e.g. a nail in a tire) and its behavior as the wheel rolls along on a flat surface, in a straight line in its own plane.

The "bouncing," curvy, trail that it leaves in the air—that's a cycloid.

A cycloid is the curve traced out by a point on the circumference of a circle as the circle rolls along on a straight line, in its own plane.

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Parametric Equations of the (basic) Cycloid Curve

$$\mathbf{ x(t) = r(\theta-\sin\theta) \quad y(t) = r(1-\cos\theta) }$$$$\text{ For } \theta \in \mathbb{R}$$

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Applications

Cycloids have many applications, including solutions to the Brachistochrone and Tautochrone problems.

Brachistochrone & Tautochrone curves, in turn, have applications in, for example:

• Pendulums
• Clock-making (pendulums)
• Surfing & Skateboard ramp design (Google it)

Demo

This cell should have a visual demonstration from Desmos of the cycloid graph as it is traced by a rolling circle.

Unfortunately, however, iPython Notebook security constraints currently prevent embedding of HTML and/or JavaScript in Markdown cells. So, in order to incorporate this content, you'll likely need to write a small Python module that interacts with the Desmos API for you.

For now, a couple of static image representations of the graph is the best I can do.