Assignment 10

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 usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention the range of t . In many applications, we think of x and y "varying with time t " or the angle of rotation that some line makes from an initial location.

Look at the graph of

x = cos (at)
y = sin (bt)

When a=b, this graph will always be a circle with radius 1.

What happens when a and b are different values?  For fun, let a = 3 and let b = -2.

If you would like to explore the graph of this equation when a and b are different values, click here!

Let's now look at the graph of

x = a cos(t)
y = b sin (t)

Let a=b and explore different values of a=b.

As you can see from the graph, we get a set of concentric circles.  The radius of each circle is determined by the value of a=b.

Now let's explore what happens when a does not equal b.

For the red ellipse below, a=3 and b=1.
For the purple ellipse below, b=1 and a=3.

Notice that when a > b, the ellipse is stretched in the x direction.  When b > a, the ellipse is stretched in the y direction.