Imagine attaching a small reflector light to the rim of a rolling bicycle wheel. As the bike rolls along flat pavement, what path does the light trace through the air?
It is not a circle or a sine wave—it is a Cycloid!
Dubbed the "Helen of Geometers" by mathematicians, the cycloid caused legendary rivalries among Galileo, Pascal, Huygens, and Newton. It possesses remarkable mathematical secrets:
The Brachistochrone (Curve of Fastest Descent): In 1696, Johann Bernoulli challenged the world to find the path that lets a frictionless marble slide between two points under gravity in the shortest possible time. An inverted cycloid is the fastest curve in the universe!
The Tautochrone (Equal Time): Pendulum clocks built with cycloidal cheeks swing with a period that is completely independent of swing amplitude.
Roberval's Area Theorem (1634): The area under one full arch of a cycloid is exactly 3 times the area of the rolling wheel:
$$\text{Area} = 3\pi R^2$$
Wren's Arc Length Theorem (1658): The total length along one arch is exactly 8 times the radius: $L = 8R$.
In this lesson, we animate a rolling wheel using gif_circle(xc, yc, r, color), gif_fillcircle(xc, yc, r, color), and gif_line(x1, y1, x2, y2, color) to trace the cycloid path frame by frame!
gif_circle(xc,yc,r,color)
Move the mouse over a dotted box for more information.
Now you try. What happens if the marker is closer to the axle, like on a bicycle pedal? See Example 1 below for Trochoids!
Type your code here:
See your results here:
The code has ???? for wheel radius R. Replace ???? with 35 and click Run to animate the rolling wheel and cycloid arch!
Notes:
Rolling Without Slipping: A wheel of radius $R$ rolling along a line moves forward by distance $s = R \cdot \theta$, where $\theta$ is the rotation angle in radians.
Notice that at the moment of contact with the ground ($\theta = 0, 2\pi$), the tracking dot's instantaneous velocity is zero! That's why tires have traction on roads.
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