Cycloid: Derivation

A: The Original Circle

Building from the homework we can get a circle starting from the bottom and moving clockwise by using:

So we can use the equations:

Click "Animate" to see the point

trace out the path described around the circle.

If we want to translate the whole circle up

unit, we can add 1 to

Check box B, then Animate.

If we want to translate the whole circle right

units, for example, we can add

Check box C, then Animate.

If we want to translate the whole circle right

units, for example, we can add

Check box D, then Animate.

Now the graph is being moved by a variable amount. Our original circle is being translated horizontally, but at the same time, the point is moving around the circle. When , the point will have gone all the way around the circle, which will now be units to the right of where it started.
Notice that is exactly the circumference of the circle.
Click “Show Circumference” and reset the animation to see how the circumference ‘unrolls’ along the x-axis.
Every time the wheel makes one full rotation, the distance it moves equals the circumference.