# 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.

## B. Vertical Translation

If we want to translate the whole circle up unit, we can add 1 to : Check box B, then Animate.

## C. Horizontal Translation by a Constant

If we want to translate the whole circle right units, for example, we can add to : Check box C, then Animate.

## D. Horizontal Translation by a Variable

If we want to translate the whole circle right units, for example, we can add to : Check box D, then Animate.

## Observations

• ﻿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.