Cycloid

By Pei-Chun Shih

A
cycloid is the curve traced by a point on the rim of a wheel rolling over
another curve like a straight line or a fixed circle. The shape of the cycloid
depends on two parameters, the radius ** r**
of the wheel and the distance

1. When r
= d, the cycloid (or the ordinary cycloid) has the following shape:

Click here to play with the
GSP file.

2. When r
> d, it is called the *curtate cycloid*, which is the path traced out by a point on the inside of the rolling
circle.

Click here to play with the
GSP file.

3. When r
< d, which is the path traced out by a point on the outside of the wheel is
called the *prolate cycloid*.

Click here to play with the
GSP file.

Next,
we are going to generate the equation of the ordinary cycloid.

LetŐs
take a look of the diagram above that a cycloid has been placed on the x-y
plane. Let the point A, with the coordinates of (x, y), be an arbitrary point
on the first blue circle c1. When time = 0, let the point A locates on the
origin and denote it as A_{0}. Also we define a parameter ** t** which is the angle that the blue
circle has moved through along the x-axis. Within a single hump,

At
time = k, the first blue circle c_{1} has moved forward to the second
blue circle c_{k}Ős location. The x term of the point A can be
expressed as the horizontal distance from the center of the circle c_{k}
to origin minus the horizontal distance, AB, of the point A to the center of
the circle c_{k}. The horizontal distance from the center of the circle
c_{k} to origin, which is the length of OO_{k}, equals the arc
AO_{k} = 2r*(t/2) = tr
according to the formula of the perimeter for a circle. Furthermore, AB = r*sin(t)
since sin(t) = AB/r. So x = tr - r sin(t) = r(t - sin(t)).

Now,
consider the vertical part, which is the y term of the point A. At time k, when
the blue circle is in the circle c_{k}Ôs position, y equals vertical
distance of the center c_{k} to origin minus the distance from the
center of the circle c_{k} to the point B. If we denote the center of
the circle c_{k} as H, then the description above can be written as y =
HO_{k} - HB = r - r*cos(t) since cos(t) = HB/r. Therefore, y = r -
r*cos(t) = r(1 - cos(t)).

The mathematical
expression of a cycloid is

x = r(t - sin(t))

y = r(1 - cos(t))

Here
I am going to use the Graphing Calculator to explore the cycloid equation
derived earlier.

First, let the radius r = 1:

As we can see from the graph, when t ranges between 0
and 2, the equation
forms a single hump just like the one we got in the previous section.
Therefore, as t increases by the multiple of 2, we got the
ordinary cycloid of r = d = 1 like the one below.

WhatŐs the difference between cycloids with different
r values, say 2, 3, or ½ ?

As we can see from above, the different values of r
allow the cycloids to expand or to shrink. However, they donŐt affect the
cycles of the cycloids. That is to say, when t is in the range of 0 and 2, each cycloid
above completes one cycle (one hump) no matter which r-value they have.

What if r is negative numbers?

It is obvious that by changing the sign of r the
cycloid can be located in any of the four quadrants.