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We discuss the basics of parametric curves.

### The idea of parametric equations

Consider an ant crawling along a flat surface like a floor of a building. Suppose we want to describe the ant’s position and the path it takes as it moves. We could introduce an origin as well as a set of $x$ and $y$ axes on the floor. We could approximate the ant as a point and so for each moment in time we could describe the ant’s position by some pair of $x$ and $y$ coordinates. As time passes, the particular path followed by the ant would describe a curve $C$ in the plane.

See the interactive below: Use the slider for $a$ to trace out the path of the ant.

Suppose we want to describe this curve. Since the ant’s path could be complicated with lots of meandering and backtracking, it is unlikely we would be able to describe $y$ as a function of $x$ or $x$ as a function of $y$. We still might be able to find an equation in $x$ and $y$ that describes the curve $C$ as simply a set of points in the plane. We call such an equation a Cartesian description of the curve since it gives the curve in terms of Cartesian coordinates $x$ and $y$. The relationship between the two coordinates could be very complicated.

Having an equation involving $x$ and $y$ would give us a static view of the curve. We see the path of the ant, all at once, like the chemical trail left behind by the ant. Although useful, there are many natural questions that we would not be able to answer using this characterization of the curve. When was the ant at a given point $p$? How fast was the ant traveling at that point? Did the ant stop at any point in time? Simply having the trail left by ant leaves out a great deal of useful information.

For each moment in time, we have some unique associated point in the plane corresponding to the ant’s position at that given time. We could describe the position of the ant as a pair of functions that depend on $t$:

These are called parametric equations for the curve $C$. In these equations we call $t$ the parameter. In many examples, the parameter will be time but the parameter can have other interpretations as well. Now given specific functions for $x(t)$ and $y(t)$ we could answer questions about the location of the ant at a given time or we could determine the velocity of the ant at a given moment. This gives us a dynamic view of the curve in that it emphasizes how the curve is traced out as the parameter changes. In fact, we can think of a parametric curve as a new kind of function where $t$ is the input and the output is a point in the plane. We won’t stress this interpretation at the moment but later in your studies you will return to this notion as your primary way of thinking of curves given parametrically.

### Visualizing parametric equations

Suppose we are are given a parametric equation algebraically, such as How can we visualize the curve traced out in $xy$-plane?

Think back to when you first learned how to graph a curve like $y=x^2$. I’m pretty sure you used a so-called “T-chart,” and if $y = x^2$, I bet it looked something like this:

Eventually you did this enough times that you just learned the basic geometric shape of such a parabola. Similarly you have memorized the basic shapes of the graph for many other functions.

Suppose now we want to graph a curve given parametrically.

With a parametric plot, both $x$ and $y$ are now functions of a third parameter, we’ll call it $t$, often thought of as time. In the same way, we can make a chart. Here $t$ is the input and $x$ and $y$ are the outputs of the two different functions $x(t)$ and $y(t)$. Graphing these points, we will get:

It’s not obvious that should have been a line. If instead we had used the parameter $s=t^3$, we could trace out the same curve as

Generally, when both $x(s)$ and $y(s)$ are lines, the curve will always be a line. Can you find the Cartesian equation of this line?

If you solve for $s$ in the $x$ equation, you can replace $s$ with an equation in terms of $x$ in the $y$ equation.

Generally graphing parametric curves are more difficult since the $x$ and $y$ values are both changing as $t$ varies and it can be difficult to see how $x$ and $y$ are related to one another. There are some curves other than lines that you should be able to visualize.

In the example of the circle above, what is the positive orientation?
clockwise counterclockwise

Using parametric equations allows us to describe many curves which would be too difficult to describe otherwise.

Do the parametric equations

as $t$ runs from $0$ to $2\pi$ define a function of $t$?

No, because the graph does not pass the vertical line test. Yes, it is a function of $t$, because for each input $t$, there is exactly one output value, an ordered pair.

### Projectile Motion

Parametric descriptions of curves occur in many applications. For example, in physics, a common scenario is to have a object acted upon by forces. We use information about the initial state of the system as well as the laws of motion to solve for how the object will behave. This description will naturally be equations which describe how the position of the object, in a coordinate system, will change with time. Therefore, parametric descriptions of curves are very natural way to describe the motion of objects. Although we will focus on describing curves parametrically in the plane, the ideas we cover can be easily generalized to three dimensions and higher.

Explore changing the initial velocities $v_{0x}$ and $v_{0y}$ in the $x$ and $y$ directions and see how that changes the trajectory of the ball by use the slider $a$ to trace the curve.

### Multiple parametrizations of a given curve

Given a curve $C$, there are many different sets of parametric equations that trace out $C$.

Previously we saw that we can parametrize the circle as

as $t$ varies over the interval $[0, 2\pi )$, traces out the circle in a counterclockwise fashion.

Suppose we replace $t$ by $-t$ in these equations.

where $t$ runs over the interval $[0, 2\pi )$. Let’s make a table.

Now the positive orientation of the curve is clockwise. Notice that if we replace $t$ by any expression in $t$, we will trace out (a portion of) the same curve, but possibly at a different speed and/or in a different direction.

Which of the following parametric equations draw the line $y=x$?
$x(t) = t$ and $y(t) = t$ for $-\infty < t < \infty$ $x(t) = t-5$ and $y(t) = t-5$ for $-\infty < t < \infty$ $x(t) = \sin (t)$ and $y(t) = \sin (t)$ for $-\infty < t < \infty$ $x(t) = \tan (t)$ and $y(t) = \tan (t)$ for $-\pi /2 < t < \pi /2$ $x(t) = t^2$ and $y(t) = t^2$ for $-\infty < t < \infty$ $x(t) = t^3$ and $y(t) = t^3$ for $-\infty < t < \infty$
Which of the following parametric equations draw the circle $(x-1)^2 + (y-2)^2 = 3^2$?
$x(t) = 1 + 3\cos (t)$ and $y(t) = 2 + 3\sin (t)$ for $0 \le t \le 2\pi$ $x(t) = 1 + 3\cos (t)$ and $y(t) = 2 + 3\sin (t)$ for $2\pi \le t \le 4\pi$ $x(t) = 1 + 3\sin (t)$ and $y(t) = 2 + 3\cos (t)$ for $0 \le t \le 2\pi$ $x(t) = 2 + 3\sin (t)$ and $y(t) = 1 + 3\cos (t)$ for $0 \le t \le 2\pi$ $x(t) = 1 + 3\cos (e^t)$ and $y(t) = 2 + 3\sin (e^t)$ for $0 \le t \le \ln (2\pi )$ $x(t) = 1 + 3\cos (e^t)$ and $y(t) = 2 + 3\sin (e^t)$ for $\ln (2\pi ) \le t \le \ln (4\pi )$

### When is it easy to find a parametrization of a curve?

Suppose you are given a curve $C$ and you want to find a parametrization for $C$. There are certain nice curves where we can easily come up with an explicit parametrization.

#### Graphs of functions

If you are given the graph of a function $y=f(x)$ then it is easy to find a parametric description of the curve. The key idea is that the independent variable is already acting like a parameter so we can use

In this way, we can easily parametrize any curve given as the graph of a function.

Can you use the technique described immediately above to express $y = e^x$ as a parametric function?

#### Lines

Suppose we are given two points $p_{1}=(a,b)$ and $p_{2}=(c, d)$. Suppose we are interested in parametrizing the line segment from $p_{1}$ to $p_{2}$. There is a canonical way to parameterize the line segment so that the parameter is $0$ at $p_{1}$ and is equal to $1$ at $p_{2}$. In a previous example, we have seen that if both $x(t)$ and $y(t)$ are lines then the curve traced out is a line. Thus it suffices to construct a line $x(t)$ so that $x(0)=a$ and $x(1)=c$. Similarly we construct a line $y(t)$ so that $y(0)=b$ and $y(1)=d$. This leads to the equations

where we let $t$ vary over the interval $[0,1]$.

If we let $t$ vary over the entire real line in this example, then we get the unique line that passes through the points $p_{1}$ and $p_{2}$.

Use this method to find a parametric equation for a line segment that starts at the point $(2,1)$ and ends at the point $(8, 5)$.

Let for $t$ in the interval $[0, 1]$.

#### Circles

The standard form for a circle centered at a point $(a,b)$ with radius $r$ is given by One problem with the standard form for a circle is that it is somewhat difficult to find points on the circle. A parametric equation representing a circle solves this problem.

One day while trying to graph a unit circle, you accidentally write down

What happens now? Do you still get a circle? How is this different from what we did in the previous question?

you still plot a unit circle in a counterclockwise fashion, with the same starting and ending points you plot a unit circle but in a clockwise fashion, with the same starting and ending points you still plot a unit circle in a counterclockwise fashion, but the starting and ending points are different you plot a unit circle but in a clockwise fashion, but the starting and ending points are different this no longer plots a circle

In mathematics, when parameterizing closed curves (like circles), the convention is to draw them in a “counterclockwise” direction. This is called the positive orientation.

What should $x(t)$ and $y(t)$ be to parameterize the circle in a counterclockwise fashion, with $t=0$ corresponding to $(1,0)$?

### Converting from parametric to cartesian

We mentioned earlier that graphing a curve from a given set of parametric equations directly from can be quite difficult. Usually computational software is needed. However there are some occasions where we can eliminate the parameter and obtain an equation involving only $x$ and $y$.

Here are some basic strategies to try:

• Solve for $t$.
• Solve for a function of $t$.
• Use a trigonometric identity.

In each case the process that we are using is called elimination of a parameter.

We’ll give several examples of how one actually eliminates a parameter.

#### Solving for the variable

In the first example, we’ll solve for $t$.

We see in this situation that the set of points traced out the curve (as we vary over all possible values of $t$) is a parabola.

#### Solving for a common function

Sometimes instead of solving for the parameter directly, it is easier to solve for a function of the parameter that is common to both $x(t)$ and $y(t)$.

#### Solving for related functions

These techniques are useful to know since they can help you more easily visualize certain simple parametric curves. However, when you eliminate the parameter it is important to realize that you are losing information about how the curve is traced out.