Parametric Equations

Typically, we draw a graph by finding a direct relationship between $x$ and $y$. Sometimes, though, it is more natural to think of $x$ and $y$ as both functions of some third variable $t$ (think time in physical applications), and then see how $x$ and $y$ vary as $t$ varies.

In other words, instead of having $y$ be a function of $x$ ($y = f(x)$), we will write $x$ and $y$ as functions of a parameter $t$: $\begin{aligned} x &= f(t)\ y &= g(t) \end{aligned}$ We can still draw a graph in the $xy$ plane; now for each value of $t$, we calculate values for $x$ and $y$ and plot the point $(x,y)=(f(t),g(t))$.

We can also eliminate the parameter to get a direct relationship between $x$ and $y$. To do so, solve for $t$ in one of the equations (if you can) and substitute this expression for $t$ into the other equation.

More Examples of Parametric Graphs

1. Parametric Circle

The following set of parametric equations describe a circle: $\begin{aligned} x &= 4 \cos 2\pi t \ \ \ \ \ \ \ \ 0 \leq t \leq 1\ y &= 4 \sin 2\pi t \end{aligned}$

Note: $\begin{aligned} x^2+y^2 &= (4\cos 2\pi t)^2 + (4\sin 2\pi t)^2\ &= 16(\cos^2 2\pi t + \sin^2 2\pi t)\ &= 16 \end{aligned}$

and $x^2+y^2=16$ is the equation of a circle centered at the origin with radius $4$.

Notice that as $t$ increases, the graph moves counterclockwise.

Also note (and this is important) that the same equation $x^2+y^2=16$ could have been written as the set of parametric equations $\begin{aligned} x &= 4 \cos t \ \ \ \ \ \ \ \ 0 \leq t \leq 2\pi\ y &= 4 \sin t \end{aligned}$

The key here is that parametrization is not unique. It is possible to parametrize a curve in different ways; one way may be easier than others, though.

2. Parametric Line

An example of a line in the $xy$ plane is written parametrically as $\begin{aligned} x &= 2+3t \ \ \ \ \ \ \ \ -\infty \leq t \leq \infty\ y &= -1-2t \end{aligned}$

Of course, to graph a line, we only need to plot two points and connect them, so the following graph shouldn't be hard to obtain:

Notice what's happening here: when $t=0$, we get a starting point (the point $(2,−1)$ in this case), and then every time $t$ increases by $1$, $x$ increases by $3$ and $y$ decreases by $2$ (the coefficients of $t$ in the parametric equations). Therefore, the slope of this line (rise over run) will be $-2/3$.

Parametric Equations of a Line

In general, the equations $\begin{aligned} x &= x_0+at \ \ \ \ \ \ \ \ -\infty < t < \infty\ y &= y_0+bt \end{aligned}$

where $a\ \cancel=\ 0$, describe a line in the $xy$ plane passing through the point $(x_0, y_0)$ with slope $\dfrac{b}{a}$.

Derivatives for Parametric Curves

We'll wrap up our discussion of parametric equations with a quick mention of how to differentiate them.

Recall the Chain Rule: $\dfrac{dy}{dt} = \dfrac{dy}{dx} \cdot \dfrac{dx}{dt}$

Therefore, $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt},\ \ \textrm{if } \dfrac{dx}{dt}\ \cancel=\ 0.$

Graphs in Polar Coordinates

Recall how polar coordinates are defined.

$\begin{aligned} x &= r \cos \theta\ y &= r \sin \theta\ \ r &= \sqrt{x^2+y^2}\ \theta &= \tan^{-1} \dfrac{y}{x} \end{aligned}$

We can use this to graph polar equations.

In general,

• $r=a$ is a circle of radius $|a|$ centered at the origin.
• $r=2a \sin θ$ is a circle of radius $|a|$ centered at $(0,a)$.
• $r=2a \cos θ$ is a circle of radius $|a|$ centered at $(a,0)$.

Graphing Polar Equations with Matlab

1. For example, to graph $r=1 + \sin θ$, follow these steps:

>> theta = 0:0.01:2*pi;
>> r = 1+sin(theta);
>> polarplot(theta,r)

2. $r = 3 \sin 2\theta$:

>> theta = 0:0.01:2*pi;
>> r = 3*sin(2*theta);
>> polarplot(theta,r)

3. $r = \cos \dfrac{2\theta}{5}, 0 \leq \theta \leq 5\pi$:

>> theta = 0:0.01:5*pi;
>> r = cos(2*theta/5);
>> polarplot(theta,r)

4. $r = \sin 2\theta\ \cos 2\theta$:

>> theta = 0:0.01:2*pi;
>> r = sin(2*theta).*cos(2*theta);    %Note the use of .* (component-wise multiplication)
>> polarplot(theta,r)