Parametric Curves

Recall that a function $f$ assigns each element in some set $A$ exactly one element in some set $B$. In certain cases, we will want to apply calculus to curves instead, which do not necessary follow the definition of a function. We can still apply rules of calculus to determine the slopes of tangents, concavity, etc…, though we will first need to familiarize ourselves with these parametric curves.

We will look at parametric curves in $\mathbb{R}^2$ (the plane) to start - and much later we will look at parametric curves in higher dimensions

Defining a Parametric Curve.

Definition: A Parametric Curve in $\mathbb{R}^2$ is a set of equations $x = x(t)$ and $y = y(t)$ that trace a curve $C$ at the Parameter $t$ varies.

It is important to note that parametric curves need not be functions - that is a parametric curve may not pass the vertical line test. Also, some authors instead define the functions $x = f(t)$ and $y = g(t)$ instead, which is perfectly fine.

So, a parametric curve is defined under two separate functions for the $x$-coordinates and $y$-coordinates of the curve under a third variable called a parameter. Often the parameter "$t$" is used and is often symbolically used to represent "time" as a particle traverses a curve. For example, consider a curve defined by the equations $x(t) = t^2$ and $y(t) = 2t$.

To graph this curve, we can first begin to determine its general shape. An easy way to do this is to get arbitrary values of $t$, and see what $(x, y)$ coordinates we obtain from those values of $t$. For example, let's test $t = -2, -1, 0, 1, 2$, as seen below:

$t$ $x(t)$ $y(t)$
-2 4 -4
-1 1 -2
0 0 0
1 1 2
2 4 4

When we plot these points, it should become rather clear what the graph approximately looks like. When using a graphing device, we obtain that the exact graph looks like:


Notice that this parametric curve looks like a parabola. In this case, we can verify that it is in fact a parabola by a process known as Eliminating The Parameter.

Since $x(t) = t^2$ and $y(t) = 2t$, we can do substitution to eliminate the parameter $t$ as follows:

\begin{align} t = \frac{y}{2} \\ x = (\frac{y}{2})^2 \end{align}

The equation for the curve above is clearly a parabola. Of course, it is not always necessarily that easy to eliminate the parameters, and in many cases the graph becomes harder to distinguish.

Note: If a parametric curve $C$ is defined on the interval $t \in [a, b]$, then $(a, f(a))$ is said to be the Initial Point Defining $C$, and $(b, f(b))$ is said to be the Terminal Point Defining $C$. This is important to distinguish when we restrict $t$ to certain values.

Example 1

Draw a graph for the parametric equation defined by: $x(t) = \sin t$ and $y(t) = cos t$.

We will begin once again with a table of values and test some arbitrary values for t to obtain some coordinates:

$t$ $x$ $y$
0 -1
-π/2 -1 0
0 0 1
π/2 1 0
π 0 -1

The points $(0, -1)$, $(-1, 0)$, $(0, 1)$, $(1, 0)$, and $(0, -1)$ aren't necessarily the most helpful points, but by computing other values of t such as $t = π/3, π/4, π/6, ...$, we can obtain more useful coordinates. It turns out that the following parametric curve is a circle, and here is how we will prove it.

Firstly, notice that $x^2 = \sin ^2 t$ and $y^2 = \cos ^2 t$, and so $x^2 + y^2 = \sin ^2 t + \cos ^2 t = 1$. Thus we have an equation of a circle which is what our parametric equations sketch out as $t$ varies.

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