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PARAMETRIC EQUATIONS

By: Lauren Lee

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A parametric curve in the plane is a pair of functions

(x = f(t) and y = g(t)), 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.

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Let’s look at the following parametric equations and vary a and b:

for 0 < t < 2pi

What happens when we set a and b equal?

Let’s look at a = 2 and b = 2

You will notice that we get a circle with radius one.

Further investigations revealed to me that this will always be the case when a = b.

Let’s see what happens when a and b aren’t equal.

We’ll let a = 2 and vary the values of b.

What happens when b = 4?

What happens when b = 6?

And when b = 10 :

Notice in these examples that the number of circular shapes created is equal to the value of b divided by the value of a.

Now let’s see what happens when we hold b constant and vary a.

Let’s make b = 2.

Let a = 4

Let a = 6

Let a = 10

These values of b seem to produce the same curves for a, except that they are rotated 90 degrees.

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Just for fun

Now let’s see an example for large values of a and b.

Here a = 19 and b = 29

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