Parametric Curves

by: Maggie Hendricks

A parametric curve in the plane is a pair of functions

    x = f(t)
    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; for this exploration, we will be looking at parametric curves for values of t between 0 and 2π. Specifically, we will be looking at the following parametric curve for these values of t:

x = a cos(t)

y = b sin(t)

To begin, let’s first look at this curve when a and b both are equal to 1. This gives us the following curve:

We see here that the result is a circle centered at the origin with a radius of 1. If we increase the value of a and b, but hold them so that they are still equal to one another, we get a similar curve (a circle centered at the origin with a larger or smaller radius). This is true even if a and b are both negative. For example, see the figure below for when a and b are both equal to -2.

We’ve now seen what happens when a is equal to b. But what happens when a and b are not equal to each other? Firstly, let’s take a look specifically at what happens where a>b. Consider the case when a is equal to 2 and b is equal to 1. See the graph below for the graph of this curve.

Notice that instead of creating a circle, this curve takes the shape of an oval, where the domain extends from -2 to +2 and the range extends from -1 to +1. It is logical to think then that perhaps the parameter a controls the domain or horizontal width of the curve while the parameter b controls the range or vertical height of the curve. To investigate further, let’s take a look at what happens when we let a be equal to 5 and let b be equal to 2. See the graph below for the graph of this curve.  

As was probably expected, this curve creates an oval where the domain extends from -5 to +5 and the range extends from -2 to +2. Is the same true when a and/or b is negative? See the graph below of the curve when a is equal to -5 and b is equal to 2.

We see that the shape of the curve is identical to the curve when a is +5, so we can assume that whether the parameters a and b are positive or negative has no real bearing on the shape of the curve. From here, we might generalize that for this particular curve, when a>b the shape of the curve is an oval whose width is greater than its height.

Now let’s take a look at what happens when a<b. As an example, see the graph of the curve for when a is equal to 2 and b is equal to 4.

Again, we probably could have predicted the shape of this graph. We see an oval shape where the domain extends from -2 to +2 and the range extends from -4 to +4. Similarly to before, we could generalize that for this curve, when a<b, the shape is an oval whose height is greater than its width. We could take this generalization a step further and say that the shape of the curve (whether circular or oval) extends at its widest a length of 2|a| and at its tallest by a length of 2|b|.

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