A **parametric curve** in the plane is a
pair of functions

where the two continuous functions define ordered pairs (x,y). The two equations are called the parametric equations of a curve.

For example, the parametric curve

looks like the graph below

for 0<=t<=1. Notice, that in order to complete the graph, we need 0<=t<=2Pi, because the period of sin(t) and cos(t) is 2Pi.

Observe, that the graph of the above parametric equation is the unit circle when we take t from 0 to 2Pi. Realize that x and y are varying with the angle of rotation t; So when t =0, we get the point (x,y)->(1,0), since cos(0)=1 and sin(0)=0, or when t=Pi/2, we get the point (0,1).

Another issue that students need to be careful with, when using technology to graph parametric equations, is the increments of t. If we do not take enough values of t between 0 and 2Pi, the graph we see can be misleading. For example, the graph above is using a small increment value, but if we graph the same parametric equation above using increments of t to be Pi/2, then we get the graph below.

Essentially, it appears that the software is plotting some points and connecting these points with line segments. In the case where the increment value of t is Pi/2, we get four points for (x,y). When t =0, Pi/2, Pi, 3Pi/2, the corresponding points are (1,0), (0,1), (-1,0), (0,-1). These points are connected by line segments, making the parametric curve appear to be a square. When the increment value of t is small enough, the connected line segments appear to be one continuous curve as in the second graph above.

Now lets consider what happens to the graph of

as we vary the parameters a and b. The graph below shows graphs of the above parametric equation with a=b=0.5, a=b=1, and a=b=2.

Notice, that if *a*=b,
the graph of the parametric equation is a circle, and the parameters
*a* and b determine the radius of the circle. The blue circle,
where a=b=1, is the unit circle. We know this because of the identity

Since our parametric curve
is determined by x* *= a cos(t) and y = b sin(t), where a=b=1,
we can substitute to obtain

What happens if *a* is
not equal to b?

Consider the parametric curves
with the constant *a* fixed at 1 and varying the parameter
b. Below are the parametric curves with b=1, b=2,
b=2.5, b=3.

Notice that when a and b are both 1, the graph
is a circle with radius 1. As b is increased, and *a *remains
fixed, the graph is an ellipse with the y-axis as the major axis.
As b increases, the ellipse is stretched vertically by a factor
of b.

We know that the curve is an ellipse when *a*
is not equal to b again because of the identity

Since and , we can substitute to obtain

This shows that the sum of the distance from a point on the curve to two foci is a constant. Thus, the curve is an ellipse.

Notice, if we graph the parametric equation and the rectangular version of the curve, with a=1 and b=2, the graphs over lap, as they should.

If b remains fixed and *a* is changed,
the curves are affected as follows: The graph below shows when
*a*=1, *a*=1.5, *a*=2, *a*=3.

Notice that when *a*>b, the graph is
an ellipse with the x-axis as the major axis.

**Observations:**

1. If a=b, then the parametric curve is a circle with radius=a=b.

2. If a>b, then the parametric curve is an ellipse with the x-axis as the major axis of symmetry.

3. If a<b, then the parametric curve is an ellipse with the y-axis as the major axis of symmetry.

4. As *a* increases, the graph is stretched
horizontally.

5. As b increases, the graph is stretched vertically.

Let's consider what happens if we perturbate the parametric equations by the vector [h sin(t), h cos(t)]', where h>0 is a real number. Then the parametric equations of the curve are defined by

Consider the graph when a=b=1 and h=1, h=.05, h=.5, and h=2.

It appears that when h=1, the curve collapses to the line y=x with domain and range [-1.41421, 1.41421]. The graph when h=.05 is almost a circle, but not quite. The curve looks slightly tilted so that the major axis is y=x. When h=.5, the curve looks like a more defined ellipse with the major axis at y=x, and when h=2, we get a larger ellipse again with y=x as the major axis.

I conjecture that if h<0 and a=b=1, then the major axis of symmetry will be y=-x. Is this the case?

The graph below shows the parametric curves

with h=1, h=.05, h=.5, and h=2.

This is just as I suspected. What happens if I complicate things even more and vary a and b?

Consider the graph with fixed h=1 and a=b=1, a=1 with b=2, and a=2 with b=1.

Notice that when a=1 and b=2, the major axis is the appears to be the line y=-8/5 x, and when a=2 and b=1 the major axis appears to be the line y=-5/8 x. I have not proven this, but below is the graphs including these lines.

There are some interesting graphs obtained by parametric equations. Here are some links to some web-sites which show cool parametric curves.

The site below allows you to plot parametric curves:

http://mss.math.vanderbilt.edu/cgi-bin/MSSAgent/~pscrooke/MSS/plotparametric2D.def

The site below has tutorials that illustrate how to draw the graph of parametric equations:

http://archives.math.utk.edu/visual.calculus/0/parametric.6/

Below is an investigation using parametric equations by Dr. Jim Wilson: