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Online Notes / Calculus II (Notes) / Parametric Equations and Polar Coordinates / Parametric Equations and Curves

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To this point (in both Calculus I and Calculus II) we’ve looked almost exclusively at functions in the form  or  and almost all of the formulas that we’ve developed require that functions be in one of these two forms.  The problem is that not all curves or equations that we’d like to look at fall easily into this form. 

 

Take, for example, a circle.  It is easy enough to write down the equation of a circle centered at the origin with radius r.

 

 

 

However, we will never be able to write the equation of a circle down as a single equation in either of the forms above.  Sure we can solve for x or y as the following two formulas show

 

 

but there are in fact two functions in each of these. Each formula gives a portion of the circle.

 

 

 

Unfortunately we usually are working on the whole circle, or simply can’t say that we’re going to be working only on one portion of it.  Even if we can narrow things down to only one of these portions the function is still often fairly unpleasant to work with.

 

There are also a great many curves out there that we can’t even write down as a single equation in terms of only x and y.  So, to deal with some of these problems we introduce parametric equations.  Instead of defining y in terms of x (  ) or x in terms of y (  ) we define both x and y in terms of a third variable called a parameter as follows,

 

 

 

This third variable is usually denoted by t (as we did here) but doesn’t have to be of course.  Sometimes we will restrict the values of t that we’ll use and at other times we won’t.  This will often be dependent on the problem and just what we are attempting to do.

 

Each value of t defines a point  that we can plot.  The collection of points that we get by letting t be all possible values is the graph of the parametric equations and is called the parametric curve.

 

Sketching a parametric curve is not always an easy thing to do.  Let’s take a look at an example to see one way of sketching a parametric curve.  This example will also illustrate why this method is usually not the best.

 

Example 1  Sketch the parametric curve for the following set of parametric equations.

                                                

Solution

At this point our only option for sketching a parametric curve is to pick values of t, plug them into the parametric equations and then plot the points.  So, let’s plug in some t’s.

 

t

x

y

-2

2

-5

-1

0

-3

 

 

-2

0

0

-1

1

2

1

 

The first question that should be asked at this point is, how did we know to use the values of t that we did, especially the third choice?  Unfortunately there is no real answer to this question.  We simply pick t’s until we are fairly confident that we’ve got a good idea of what the curve looks like.  It is this problem with picking “good” values of t that make this method of sketching parametric curves one of the poorer choices.  Sometimes we have no choice, but if we do have a choice we should avoid it.  We’ll discuss an alternate graphing method in later examples.

 

We have one more idea to discuss before we actually sketch the curve.  Parametric curves have a direction of motion.  The direction of motion is given by increasing t.  So, when plotting parametric curves we also include arrows that show the direction of motion.  We will often give the value of t that gave specific points on the graph as well to make it clear the value of t that gave that particular point.

 

Here is the sketch of this parametric curve.

ParaEqn_Ex1_G1

 

So, it looks like we have a parabola that opens to the right. 

 

Before we end this example there is a somewhat important and subtle point that we need to discuss first.  Notice that we made sure to include a portion of the sketch to the right of the points corresponding to  and  to indicate that there are portions of the sketch there.  Had we simply stopped the sketch at those points we are indicating that there was no portion of the curve to the right of those points and there clearly will be.  We just didn’t compute any of those points.

 

This may seem like an unimportant point, but as we’ll see in the next example it’s more important than we might think at this point.

 

Before addressing a much easier way to sketch this graph let’s first address the issue of limits on the parameter.  In the previous example we didn’t have any limits on the parameter.  Without limit on the parameter the graph will continue in both directions as shown in the sketch above.

 

We will often have limits on the parameter however and this will affect the sketch of the parametric equations.  To see this effect let’s look a slight variation of the previous example.

 

Example 2  Sketch the parametric curve for the following set of parametric equations.

                              

Solution

Note that the only difference here is the presence of the limits on t.  All these limits do is tell us that we can’t take any value of t outside of this range.  Therefore, the parametric curve will only be a portion of the curve above.  Here is the parametric curve for this example.

ParaEqn_Ex2_G1

 

Notice that with this sketch we started and stopped the sketch right on the points originating from the end points of the range of t’s.  Contrast this with the sketch in the previous example where we had a portion of the sketch to the right of the “start” and “end” points that we computed.

 

In this case the curve starts at  and ends at , whereas in the previous example the curve didn’t really start at the right most points that we computed.  We need to be clear in our sketches if the curve starts/ends right at a point, or if that point was simply the first/last one that we computed.

 

It is now time to take a look at an easier method of sketching this parametric curve.  This method uses the fact that in many, but not all, cases we can actually eliminate the parameter from the parametric equations and get a function involving only x and y.  There will be two small problems with this method, but it will be easy to address those problems.  It is important to note however that we won’t always be able to do this.

 

Just how we eliminate the parameter will depend upon the parametric equations that we’ve got.  Let’s see how to eliminate the parameter for the set of parametric equations that we’ve been working with to this point.

 

 

Example 3  Eliminate the parameter from the following set of parametric equations.

                                                

Solution

One of the easiest ways to eliminate the parameter is to simply solve one of the equations for the parameter (t, in this case) and substitute that into the other equation.  Note that while this may be the easiest to eliminate the parameter, it’s usually not the best way as we’ll see soon enough.

 

In this case we can easily solve y for t.

                                                                

 

Plugging this into the equation for x gives,

                                         

 

Sure enough from our Algebra knowledge we can see that this is a parabola that opens to the right.

 

We won’t bother with a sketch for this one as we’ve already sketched this once and the point here was more to eliminate the parameter anyway.

 

Getting a sketch of the parametric curve once we’ve eliminated the parameter is fairly simple.  All we need to do is graph the equation that we found by eliminating the parameter. 

 

As noted already however, there are two small problems with this method.  The first is direction of motion.  The equation involving only x and y will NOT give the direction of motion of the parametric curve.  This is an easy problem to fix however.  All we need to do is plug in some values of t into the parametric equations and we can determine direction of motion from that.  How many values of t we plug in will depend upon the parametric equations.  In some cases only two will be required and in others we might need more points.

 

The second problem is best illustrated in an example as we’ll be running into this problem in the remaining examples.

 

Example 4  Sketch the parametric curve for the following set of parametric equations.  Clearly indicate direction of motion.

                             

Solution

In this case we could eliminate the parameter as we did in the previous section by solving one of these for t and plugging this into the other.  For example,

                          

 

Can you see the problem with doing this?  This is definitely easy to do but we have a greater chance of correctly graphing the original parametric equations than we do graphing this!

 

There are many ways to eliminate the parameter from the parametric equations and solving for t is usually not the best way to do it.  While it is often easy to do we will, in most cases, end up with an equation that is almost impossible to deal with.

 

So, how can we eliminate the parameter here?  In this case all we need to do is recall a very nice trig identity and the equation of an ellipse.  Let’s notice that we could do the following here.

                                    

 

Eliminating the middle steps gives us,

                                                                

and so it looks like we’ve got an ellipse.

 

Before proceeding with this example it should be noted that what we did was probably not all that obvious to most.  However, once it’s been done it does clearly work and so it’s a nice idea that we can use to eliminate the parameter from some parametric equations involving sines and cosines.  It won’t always work and sometimes it will take a lot more manipulation of things than we did here.

 

An alternate method that we could have used here was to solve the two parametric equations for sine and cosine as follows,

                                                

 

Then, recall the trig identity we used above and these new equation we get,

                                        

 

So, the same answer as the other method.  Which method you use will probably depend on which you find easier to use.  Both are perfectly valid and will get the same result.

 

Now, let’s continue on with the example.  We’ve identified that the parametric equations describe an ellipse, but we can’t just sketch an ellipse and be done with it.  Recall that all parametric curves have a direction of motion. 

 

So, we next need to determine the direction of motion.  The equation of the ellipse tells us nothing about the direction of motion.  To get the direction of motion we’ll need to go back to the parametric equations and plug in a few points.  Note as well that in this case we’ll need more than two points to do this.  Given any two points on an ellipse we could get between them by going either clockwise or counter-clockwise about the circle.  So, we’ll need at least three points to accurately determine the direction of motion.

 

While doing this we should also keep in mind that we’ve been given a range of t’s to work with and as we saw in Example 2 this may mean that we will only get a portion of the actual ellipse.  So, let’s choose t’s that will cover the whole range.  This will give us the direction of motion and enough information to determine what portion of the ellipse is in fact traced out.

 

Note that this is the second problem alluded to above in eliminating the parameter.  Once we have eliminated the parameter we’ve not only eliminated the direction of motion, but we’ve also eliminated any information about what portion of the actual graph is traced out by the parametric equations.  We will always need to keep in mind that this a potential problem when eliminating the parameter from parametric equations.

 

So, here is a table of values for this set of parametric equations.

 

t

x

y

0

5

0

 

0

2

 

-5

0

 

0

-2

 

5

0

 

It looks like we are moving in a counter-clockwise direction about the ellipse and it also looks like we’ll make exactly one complete trace of the ellipse in the range given.

 

Here is a sketch of the parametric curve.

ParaEqn_Ex4_G1

 

Let’s take a look at another example.

 

Example 5  Sketch the parametric curve for the following set of parametric equations.  Clearly indicate direction of motion.

                             

Solution

Note that the only difference in these parametric equations is that we replaced the t with 3t.  We can eliminate the parameter here using either of the methods we discussed in the previous example.  In this case we’ll do the following,

                               

 

So, we get the same ellipse that we did in the previous example.  However, we also don’t get the same parametric curve in some sense.  We saw in the previous example that we make one complete trace of the ellipse in the range .  In this set of parametric curves we don’t have just a t in the trig functions however.  In this set we’ve got a 3t.

 

When we have a t we know that we’ll complete a single trace when  so to determine a t that will complete a single trace when we have a 3t all we need to do is,

                                      

 

So, while we have the same ellipse that we got in the previous example we’ll trace out the curve exactly once in the range,

                                                                 

 

Since we are working on the range  it then looks like the ellipse in this case will be traced out three times instead of only once as we got in the previous example.  Each ellipse will be traced out in the following ranges,

                           

 

The last issue we need to deal with prior to sketching is the direction of motion.  By picking values of t we can see that the direction of motion isn’t changed in this case.  However, because we’re going around faster than before we should probably use a different set this time to make sure we get an accurate idea of the direction of motion.

 

t

x

y

0

5

0

 

0

2

 

-5

0

 

Here’s the sketch and note that it really isn’t all that different from the previous sketch.  The only differences are the value of t and the various points we included.

ParaEqn_Ex5_G1

 

So, we saw in the last two examples two sets of parametric equations that in some way gave the same graph.  Yet, because they traced out the graph a different number of times we really do need to think of them as different parametric curves.  This may seem like a difference that we don’t need to worry about, but as we will see in later sections this can be a very important difference.  In some of the later sections we are going to need a curve that is traced out exactly once.

 

Let’s take a look at a couple more examples.

 

Example 6  Sketch the parametric curve for the following set of parametric equations.  Clearly identify the direction of motion.  If the curve is traced out more than once give a range of the parameter for which the curve will trace out exactly once.

                                               

Solution

We can eliminate the parameter much as we did in the previous two examples.  However, we’ll need to note that the x already contains a  and so we won’t need to square the x.  We will however, need to square the y as we need in the previous two examples.

                       

In this case we get a parabola that opens to the left. 

 

We will need to be very, very careful however in sketching this parametric curve.  We will NOT get the whole parabola.  A sketch of a parabola, in this form, will exist for all possible values of y.  However, we have defined both x and y in terms of sine and cosine and we know that the value of these are limited and so we won’t get all possible values of x and y here.  To see what values of x and y we get let’s note the following,

 

                        

 

So, it is clear from this that we will only get the portion of the parabola that is defined by the equation above.  Before sketching let’s also get the direction of motion.  Here are some points for a range of t’s.

t

x

y

0

0

2

 

1

0

 

0

-2

 

1

0

 

0

2

 

In the range  we start at (0,2) and end up back at that same point.  Recalling that we must travel along the parabola this means that we must retrace our path to get back to the starting point.  So, it looks like we’ve got a parametric curve that is traced out over and over in both directions and we will trace out once in the range .

 

ParaEqn_Ex6_G1

 

To this point we’ve seen examples that would trace out the complete graph that we got by eliminating the parameter if we took a large enough range of t’s.  However, in the previous example we’ve now seen that this will not always be the case.  It is more than possible to have a set of parametric equations which will continuously trace out just a portion of the curve.  We can usually determine if this will happen by looking for limits on x and y that are imposed up us by the parametric equation.

 

We will often use parametric equations to describe the path of an object or particle.  Let’s take a look at an example of that.

 

Example 7  The path of a particle is given by the following set of parametric equations.

                                          

Completely describe the path of this particle.  Do this by sketching the path, determining limits on x and y and giving a range of t’s for which the path will be traced out exactly once (provide it traces out more than once of course).

 

Solution

Eliminating the parameter this time will be a little different.  We only have cosines this time and we’ll use that to our advantage.  We can solve the x equation for cosine and plug that into the equation for y.  This gives,

                                       

 

This time we’ve got a parabola that opens upward.  We also have the following limits on x and y.

                       

 

So, again we only trace out a portion of the curve.  Here’s a set of evaluations so we can determine a range of t’s for one trace of the curve.

 

 

t

x

y

0

3

2

 

0

1

 

-3

2

 

0

1

 

3

2

 

So, it looks like the particle, again, will continuously trace out this portion of the curve and will make one trace in the range .  Here is a sketch of the particle's path with a few value of t on it.

ParaEqn_Ex7_G1

 

We should give a small warning at this point.  Because of the ideas involved in them we concentrated on parametric curves that retraced portions of the curve more than once.  Do not, however, get too locked into the idea that this will always happen.  Many, if not most parametric curves will only trace out once.  The first one we looked at is a good example of this.  That parametric curve will never repeat any portion of itself.

 

There is one final topic to be discussed in this section before moving on.  So far we’ve started with parametric equations and eliminated the parameter to determine the parametric curve.

 

However, there are times in which we want to go the other way.  Given a function or equation we might want to write down a set of parametric equations for it.  In these cases we say that we parameterize the function.

 

If we take Examples 4 and 5 as examples we can do this for ellipses (and hence circles).  Given the ellipse

 

 

a set of parametric equations for it would be,

 

 

This set of parametric equations will trace out the ellipse starting at the point  and will trace in a counter-clockwise direction and will trace out exactly once in the range .  This is a fairly important set of parametric equations as it used continually in some subjects with dealing with ellipses and/or circles.

 

Every curve can be parameterized in more than one way.  Any of the following will also parameterize the same ellipse.

 

 

 

The presence of the  will change the speed that the ellipse rotates as we saw in Example 5.  Note as well that the last two will trace out ellipses with a clockwise direction of motion (you might want to verify this).  Also note that they won’t all start at the same place (if we think of  as the starting point that is).

 

There are many more parameterizations of an ellipse of course, but you get the idea.  It is important to remember that each parameterization will trace out the curve once with a potentially different range of t’s.  Each parameterization may rotate with different directions of motion and may start at different points. 

 

You may find that you need a parameterization of an ellipse that starts at a particular place and has a particular direction of motion and so you now know that with some work you can write down a set of parametric equations that will give you the behavior that you’re after.

 

Now, let’s write down a couple of other important parameterizations and all the comments about direction of motion, starting point, and range of t’s for one trace (if applicable) are still true.

 

First, because a circle is nothing more than a special case of an ellipse we can use the parameterization of an ellipse to get the parametric equations for a circle centered at the origin of radius r as well.  One possible way to parameterize a circle is,

 

 

 

 

Finally, even though there may not seem to be any reason to, we can also parameterize functions in the form  or .  In these cases we parameterize them in the following way,

 

 

 

At this point it may not seem all that useful to do a parameterization of a function like this, but there are many instances where it will actually be easier, or it may even be required, to work with the parameterization instead of the function itself.  Unfortunately, almost all of these instances occur in a Calculus III course.


Online Notes / Calculus II (Notes) / Parametric Equations and Polar Coordinates / Parametric Equations and Curves

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