Thomas Earl Ricks
Assignment # 10
Assignment #10: A cycloid is the locus of a point on a circle that rolls along a line. Write parametric equations for the cycloid and graph it. Consider also a GSP construction of the cycloid.
First, we will consider constructing the cycloid on GSP, and then we will attempt to create a parametric equation for the cycloid.
We start by constructing the line that our circle will roll on:
We now wish to construct the path our circle will follow as it rolls on this line. To do this, we construct a point A on the line with a perpendicular line through A. This will serve as the uniform height that our path will pass through. We then construct a circle with the radius of our eventual rolling circle, to find a point for our path to pass through.
Now we construct the path through the upper intersection point of the perpendicular line and the circle. Note that this path is parallel to our original line. Our rolling circle will follow this path. It is a dotted red line in the diagram below:
Now we place point C on the dotted red path, which will be the center of our rolling circle, with a perpendicular line through point C, so we can create a circle that has as its radius the distance between the path and the original line. We now construct rolling circle, the solid green circle pictured below, tangent to the original line at point t:
We can now hide everything but the original line and our green circle with its center point C:
Let us add a spoke to show us the green circle rolling, by creating a segment from point C to the green circle. And let us move our circle over to the left, like so:
We now select the center point and our outer spoke point, click on “Edit” in the Menu Bar, select “Action Buttons” and then “Animate”. We wish for the circle to move “backward”, which will move it to the right, so we select that, as well as have our spoke point move in a “clock-wise” direction, so we adjust that. We are now ready to animate our rolling wheel. We select on the outer spoke point, and select “Display” and then “Trace Point”. When we push the “Animate Points” button on the screen, we should have our rolling circle roll to the right, with the spoke turning, leaving a red trace of its path on the screen like so:
This is the locus of points created by a circle rolling on path. It is actually just the beginning of the locus, of course, the first hump of an infinite chain.
Hopefully you succeeded in creating one!
If we zoom out, we will see the cycloid path in more detail, and observe that it is periodic, meaning it repeats itself over and over. To zoom out, we could unhide everything, resize our original dotted green circle to be smaller, and then re-hide everything but the line, circle and spoke. Then when we animate, we obtain:
There are all sorts of fun ideas that can be pursued from this idea of a circle rolling on a line.
How about a circle rolling along a circle? What shape would that produce? Here is one if the green circle has a diameter of half the blue.
Here is one with less than one fourth diameter, after several revolutions. What would it look like if it kept going?
How about a circle rolling outside a triangle, or a square, or a hexagon? Can you think of any others and try them? What about rolling inside these shapes?
Now we will develop the equation of a cycloid. This is not an easy task and requires some effort of thought. We can look at our GSP trace to infer some ideas about what the parametric equation will look like.
Since our rolling circle has radius r, and therefore circumference , the circle makes one complete revolution as its center travels .
Since the circle is rolling in time, let us say that it takes seconds for the circle to complete one complete revolution. By connecting time to distance rolled, it will be easier to think about the situation.
So time is our independent variable, and we wish to construct an equation that graphs the cycloid as a function of time t. We now need to observe the relative position of the spoke point as time passes, both in the x and the y directions. Let us look at the y direction first as time passes:
The spoke points are yellow to stand out from the red path, and we have added the circle rolling in two more positions. Observe that at time 0, the height of the spoke point is 0, while at
the height is 2r. Then the height falls back to r when time is at
and finally after one revolution, the height is back to 0. This process repeats itself with every revolution. The height also appears to be oscillating around the height r. It begins below the line y = r, and then passes through the line, and finally dips below the line again.
How could we describe the motion of the spoke point as a function of t?
We observe that its motion is periodic with respect to t, and that every units of time, it begins to repeat itself. What functions do this? Sine and cosine are obvious candidates.
Could we describe the height of the spoke point as a function of t with sine or cosine? Cosine would work, since if we combined it with the height r, we could get it to oscillate back and forth around the line y = r, just as the spoke point does.
We note that for time t = 0, cos(0) = 1. Thus we could find the height of the spoke point using a function of cosine by taking the height r and subtracting off of it rcos(0) = r. This would reduce our spoke point height to 0. Thus we have the equation y = r – r cos(t) as our equation.
We observe that for the other values of t, this equation accurately describes the height of the spoke point. We can see this more clearly by observing the superimposed graph y=r-rcos(t) on our rolling cycloid.
Note that the yellow spoke point doesn’t follow the light green cosine graph, but that the height of the spoke point follows the light green cosine graph.
For example, at time t =
the cosine graph has height r, while the spoke does as well.
We now need to create an equation for the horizontal or x position of the cycloid, based on t. We follow the same reasoning, but instead of looking at the height of the spoke point at time t, we observe its horizontal position, relative to the center of the rolling circle.
Observe that at time t = 0, the spoke point is also 0, but as the center advances to
the spoke point is actually r behind! It then catches up to the spoke point at
surpasses it at
and slows down to be at the same horizontal position at .
What function of t could give this result?
A little playing reveals x = r t - r sin(t) matches our values.
Thus our equations are:
which yields the cycloid:
Or if we wish to zoom further out to see its periodic behavior, we see:
What other related equations can you discover?
Click here to learn more about the Cycloid: