# Cycloid: Derivation

- Author:
- Brian Sterr

## 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.

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## 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.