We discuss the basics of parametric curves.
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 and 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 and coordinates. As time passes, the particular path followed by the ant would describe a curve in the plane.
See the interactive below: Use the slider for 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 as a function of or as a function of . We still might be able to find an equation in and that describes the curve 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 and . The relationship between the two coordinates could be very complicated.
Having an equation involving and 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 ? 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 :
These are called parametric equations for the curve . In these equations we call the parameter. In many examples, the parameter will be time but the parameter can have other interpretations as well. Now given specific functions for and 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 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.
Suppose we are are given a parametric equation algebraically, such as How can we visualize the curve traced out in -plane?
Think back to when you first learned how to graph a curve like . I’m pretty sure you used a so-called “T-chart,” and if , 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 and are now functions of a third parameter, we’ll call it , often thought of as time. In the same way, we can make a chart. Here is the input and and are the outputs of the two different functions and . Graphing these points, we will get:
It’s not obvious that should have been a line. If instead we had used the parameter , we could trace out the same curve as
Generally, when both and are lines, the curve will always be a line. Can you find the Cartesian equation of this line?
Generally graphing parametric curves are more difficult since the and values are both changing as varies and it can be difficult to see how and are related to one another. There are some curves other than lines that you should be able to visualize.
Although the parameter is , it can be helpful here to think of it as an angle. The angle is initially and you increase the angle from to . You should then recognize that these equations just give the and coordinate of a point on the unit circle. We can imagine these parametric equations as “drawing” the unit circle as changes
Once the parameter has reached , the circle has been traced out exactly once.
Using parametric equations allows us to describe many curves which would be too difficult to describe otherwise.
As the parameter increases, the curve rotates like it will trace out the circle. However instead of having a fixed amplitude that describes how far the circle is from the origin, the factor in front of the cosine and sine term can be thought of as a varying radius that increases as increases. Thus as the parameter increases, the curve rotates in a counterclockwise fashion like the circle but with a radius that keeps getting larger and larger. This leads to the following spiral shape.
as runs from to define a function of ?
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.
In differential calculus, you likely discussed projectile motion in one dimension. For example, suppose you launch a ball straight up into the air. You then want to find the position of the object with respect to time. Using integration and the fact that the ball has a constant acceleration with respect to gravity, we can find the trajectory of the ball.
Now the acceleration due to gravity is constant and we take . Call the initial velocity (this is the initial velocity in direction). Then we can integrate and use the initial velocity to find the velocity
Suppose it is launched from a height . Integrating again and using the initial position, we obtain
Now suppose we want to launch the ball at an angle instead of straight up. The ball’s path will have both a vertical and horizontal component, so this is now a two dimensional problem. We launch our ball with a certain initial speed . We again choose a coordinate system. Choose the origin at a fixed point on the ground level, choose the -axis oriented upward, and choose our -axis to be the direction in which we launch our projectile. Ignoring air resistance, the trajectory of our object will be confined to this -plane. Suppose our object is launched from an initial position in our coordinate system. Can we describe the motion of the ball?
It turns out the motion of the object can be analyzed in the and directions separately. We can describe the position of the ball as
The angle that we launch our ball determines an the initial velocity in each components. Take to be the initial velocity along the direction and to be the initial velocity along the direction. In the direction, our ball will behave just like a ball thrown straight upwards. Using our previous work on one dimensional projectile motion, we obtain
Now we want to understand the behavior in the direction. There are no forces in the direction so the acceleration is . Using the initial velocity in the direction and integrating the acceleration , we find . Integrating again and using the initial position , we obtain :
This gives us the parametric description of the trajectory of the ball in the plane with respect to time.
The curve that the projectile follows is parabolic. However the parametric description gives us more information. It gives us a dynamic description that tells us the position of the particle at each moment of time.
Explore changing the initial velocities and in the and directions and see how that changes the trajectory of the ball by use the slider to trace the curve.
Given a curve , there are many different sets of parametric equations that trace out .
Previously we saw that we can parametrize the circle as
as varies over the interval , traces out the circle in a counterclockwise fashion.
Suppose we replace by in these equations.
where runs over the interval . Let’s make a table.
Now the positive orientation of the curve is clockwise. Notice that if we replace by any expression in , we will trace out (a portion of) the same curve, but possibly at a different speed and/or in a different direction.
Suppose you are given a curve and you want to find a parametrization for . There are certain nice curves where we can easily come up with an explicit parametrization.
If you are given the graph of a function 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.
Suppose we are given two points and . Suppose we are interested in parametrizing the line segment from to . There is a canonical way to parameterize the line segment so that the parameter is at and is equal to at . In a previous example, we have seen that if both and are lines then the curve traced out is a line. Thus it suffices to construct a line so that and . Similarly we construct a line so that and . This leads to the equations
where we let vary over the interval .
If we let vary over the entire real line in this example, then we get the unique line that passes through the points and .
Let for in the interval .
The standard form for a circle centered at a point with radius 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.
In mathematics, when parameterizing closed curves (like circles), the convention is to draw them in a “counterclockwise” direction. This is called the positive orientation.
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 and .
Here are some basic strategies to try:
- Solve for .
- Solve for a function of .
- 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.
In the first example, we’ll solve for .
We see in this situation that the set of points traced out the curve (as we vary over all possible values of ) is a parabola.
Sometimes instead of solving for the parameter directly, it is easier to solve for a function of the parameter that is common to both and .
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.