Cristina Aurrecoechea Fall 2005 |

A cycloid is the locus of a point on a circle that rolls along a line. Here we present the parametric equations for the cycloid: we explain how to obtain the equations and graph the cycloid. Although the homework was asking for a GSP construction, we believe that our Graphic Calculator movie is just as interesting.

Figure 1 shows the cycloid for a rolling circle
of **radius a = 1**. This line reflects the coordinates of
a point **P** in the circumference of the rolling circle. This
point is located at (0,0) initially, at the base of the circle.
Some time later (exactly in the middle of each cycle) it will
be located at the top of the circle, with a maximum y value: **y
= 2a**; the x distance traveled by this point in each cycle
is precisely the length of the circumference, which for a = 1
is .

Here you can see the equations of a rolling circle (it rolls on the x axis).

Figure 2 is a detailed snapshot of our rolling
circle of radius **a = 1**. We will deduct the parametric equations
for the cycloid based on this Figure.

The rolling circle at time zero is shown in
purple. We look at the coordinates **(x,y)** of the point at
its base located initially at coordinates (0,0). As the circle
rolls we follow the trace of this point **(x,y)** up to the
position shown in the green circle. At this point the circle has
rotated **t radians. **The angle **t** and its arc are shown
by shading the area in light blue. The arc is equal to the product
( angle * radius ); in general for a **radius a **the arc is
**at.**

Lets calculate the coordinates **(x,y)**
of our point as a function of the angle **t. **That will give
us the parametric equations for the cycloid. Our calculations
will be based on the red triangle.

**x** is the rolled
arc (**at**) minus the horizontal side of the red triangle,
which is **a.sin (-t)**. Therefore given that **sin(-t) =
sin(t)** we have:

**y** is the radius
**a** plus the vertical side of the red triangle, which is
**a.cos(-t)**. Therefore given that **cos(-t) = -cos(t)**,
we have:

Next we have a set of movies for you to play with the cycloid function and rolling circle:

- cycloid: slider n allows you to control the number of cycles; the slider a modifies the radius.
- rolling circle,
**radius = 1,**one cycle (360 degrees): here - rolling circle,
**radius = 1**, three cycles (t=about 20 radians): here - rolling circle,
**radius = a**,**a**is provided as a slider: here

Return to Cristina's page with all the assignments.