The Cycloid

by

Susan Sexton

The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid. This curve is the locus of the P as it moves along the line.

GSP Animation

It may be better to describe a curve in the plane using parametric equations. Here the x- and y-coordinates of points on the curve are separate functions of a new variable, t, called the parameter. These parametric equations are given by:

x = f(t) and y = g(t)

The curve that is traced out by the movement of P located at (x, y) = (f(t), g(t)) is called a parametric curve.

The cycloid is a parametric curve.

What are the parametric equations for the cycloid?

Let’s look at a part of the curve created by a circle that has rolled along the line.

To find the position of D (or its x- and y-coordinates) we must look at the distances created.

The x-coordinate is the length of OT minus the length of DB.

The y-coordinate is the length of CT minus the length of CB.

The length of an arc that is subtended by a central angle is equal to the radius times the central angle. Since the distance that circle C has rolled must be equal to the length of arc DT and arc DT is equal to aq then aq = arc DT = OT.

By trigonometry we know that DB = asinq and CB = acosq.

Therefore the coordinates of D are:

x-coordinate: OT – DB = aq – asinq = a(q – sinq)

y-coordinate: CT – CB = a – acosq = a(1 – cosq)

If we put these equations in Graphing Calculator we can explore what happens when “a” varies.

However, instead of q, t will be used.

Let’s first graph a cycloid whose radius is 1.

t = p

t = 2p

t = 4p

Now let us see what happens when we vary a.

What happens when a is negative?

As stated earlier, the cycloid occurs as a

circle rolls along a line.

What happens when the circle rolls along

something else . . . like another circle?

The parametric curve traced here

is called an epicycloid.

GSP Animation

What happens when the circle rolls along the inside of a circle?

The parametric curve traced here

is called an hypocycloid.

GSP Animation

Earlier, we found the following for a general cycloid:

x-coordinate: aq – asinq

y-coordinate: a – cosq

But there are other special type of cycloids whose coordinates are the following:

x-coordinate: aq – bsinq

y-coordinate: a – bcosq

Let’s use Graphing Calculator to see what happens when we vary “a” and “b”.

First I will vary “t’.

a = 1, b = 2, t = p

a = 1, b = 2, t = 2p

a = 1, b = 2, t = 4p

Now let’s fix “a” and “t” and vary “b”.

We can see that “b” is enlarging the loop size.

This type of curve is called a

prolate cycloid and occurs when b > a.

Now let’s fix “b” and “t” and vary “a”.

A couple of things to notice here:

Where did the loops go?

The larger it gets, “a”

appears to stretch the curve.

This type of curve is called a

curtate cycloid and occurs when b < a.

Discussion:

A couple of parametric thoughts to think about –

Obviously things occurred as “a”, “b”, and “t” varied, but why?

There are many other special parametric curves that exist. Can you find them?

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