Instructor's
Notes:
Background/suggested prerequisite
activities.
The path of a particle moving in a plane need
not trace out the graph of a function, hence we cannot describe the path
by expressing y directly in terms of x. An alternate way
to describe the path of the particle is to express the coordinates of its
points as functions of a third variable using a pair of equations
x = f(t),
y = g(t).
Equations of this form are called parametric
equations for x and y, and the unknown t is called
a parameter. The parameter t may represent time in some instances,
an angle in other situations, or the distance a particle has traveled along
the path from a designated starting point.
An easy example of a parametric representation
of a curve is obtained by using basic trigonometry to obtain parametric
equations of a circle of radius 1 centered at the origin. We have the relationships
between a point (x,y) on the circle and an angle t as shown
in the following figure.
By elementary trigonometry we have the parametric
equations
x = cos(t),
y = sin(t).
As t goes from 0 to 2p
the corresponding points trace out the circle in a counter clockwise direction.
The following animation illustrates the process.
Generating a circle
parametrically.
(For a related demonstration for generation
of sine and cosine curves see the Circular Functions demo. )
Parametric equations
based on circles.
The Cycloid:
A famous curve that was named by Galileo
in 1599 is called a cycloid. A cycloid is the path traced out by
a point on the circumference of a circle as the circle rolls (without slipping)
along a straight line. A cycloid can be drawn by a pencil (chalk or marker)
attached to a circular lid which is rolled along a ruler. The following
animation illustrates the generation of a cycloid.
If the circle that is rolled has radius
a,
then the parametric equations of the cycloid are
x = a(t - sin(t)),
y = a(1 - cos(t))
where parameter t is the angle
through which the circle was rolled. As in the case of the circle, these
parametric equations can be derived using elementary trigonometry. To see
the basics of the derivation click on the following: The
Equations of a Cycloid.
For some history related to cycloids click
on the following: St.Andrews-cycloid
information.
The Epicycloid:
Take a large circle centered at the
origin. Place a smaller circle tangent to the original circle at
the point where it crosses the positive x-axis and outside of the original
circle. Identify the point of tangency. See the next figure.
Next we roll the smaller circle around the
larger circle and follow the path of the point of tangency. The resulting
curve is called an epicycloid. The shape of the curve generated
in this manner depends on the relationship between the radius of the large
circle and the small circle. The following animations illustrate three
particular cases.
 |
 |
 |
Cardioid
|
Nephroid
|
Ranunculoid |
With a careful analysis we can show that
the parametric equations of an epicycloid using a large circle of radius
a
and a small circle of radius b, where a > b, are
x=(a+b)cos(t)-bcos((a+b)t/b)
, y=(a+b)sin(t)-bsin((a+b)t/b) .
The epicycloid has been studied by such luminaries
as Leibniz, Euler, Halley, Newton and the Bernoullis. The epicycloid curve
is of special interest to astronomers and the design of cog-wheels with
minimum friction. To experiment with epicycloids see the files available
at the end of this module. For more information and an on-line animator
click on the following link:
On-line
animator for epicycloid.
The hypocycloid:
Here take a large circle centered at the
origin. Place a smaller circle tangent to the original circle at the point
where it crosses the positive x-axis and inside the original circle.
Identify the point of tangency. See the next figure.
Roll the smaller circle around the larger
circle and follow the path of the point of tangency. The resulting curve
is called an hypocycloid. The shape of the curve generated in this
manner depends on the relationship between the radius of the large circle
and the small circle. The following two animations illustrate the generation
of hypocycloids. (Also see the animation at the beginning of this demo.)
Again with a careful analysis we can show
that the parametric equations of an epicycloid using a large circle of
radius a and a small circle of radius b, where a > b,
are
x=(a-b)cos(t)+bcos((a-b)t/b)
, y=(a-b)sin(t)-bsin((a-b)t/b) .
Comments:
I have used demos of this type in several
ways.
-
In calculus class we define parametric equations
and have the students plot a few by hand before we do the demonstration.
The demonstration then wows them as they realize how difficult it would
be to plot these objects by hand. For the epicycloid I usually use
two rolls of tape with different radii, identify a point on the outer one
with a magic marker, and then roll it around the other and ask the students
what they think the path would look like. Then using DERIVE (see
downloads available below) we plot epicycloids for various radii
and ask questions about what they would expect and (as I already pointed
out in discussion) the numbers of times we need to rotate (i.e. what the
parameters should be) in order to obtain a closed epicycloid curve.
We try to get them to state a little theorem about this.
Each semester for calculus we have four computer labs. Our
students must go to the lab, perform the exercises or experiments on
the computer using DERIVE, and then write up their results. One of
these labs requires students to plot a collection of functions using
parametric equations and polar coordinates. The main purpose of
the lab is to familiarize students with the parametric plotting
capabilities of DERIVE. This lab is assigned after students have
seen some of the demo material shown above.
-
I have also used this demo or a variation
thereof during various admission (student recruiting) days.
Generally, here, the audience consists of prospective students and their
parents. It is relative easy to explain what the cycloids are and
then it is exciting and informative to see them plotted.
Downloads available:
For DERIVE the following file was supplied
by Anthony Berard and can be downloaded by clicking on
Epi-HypoCycloids.mth.
(See the imbedded instructions concerning change of scale.)
For Matlab the following files were written
for this demo and can be downloaded by clicking on the file name: epicycloid.m
, hypocycloid.m .
Credits:
This demo was submitted by
Anthony
Berard
Department of Mathematics
Kings College
and is included in Demos
with Positive Impact with his permission.
