We all know how to graph something like y=2x+3. We choose different x's, solve for the respective y's, plot the points and then try to connect them in a smooth manner. But what happens when we try to plot something much more complicated such as ? Its not so easy to plot using our guess, plug and chug method. This is one example of when we would use parametric equations, which is what we are going to be exploring in this section. Parametric equations are a tool that can be used to graph and analyze equations that would be very difficult using our traditional rectangular coordinates. We will also be able to solve a very interesting problem using parametric equations that would be much more difficult to do using more traditional methods.
Imagine for a minute a bug crawling on a table where you have drawn coordinate axes. As time progresses the bug can crawl anywhere on the table it wishes and if you read off its x and y coordinates every second you might find that its x-coordinate is a function of time as is its y-coordinate. Since its position at any given time, which we can call P(t) where t represents amount of time that has passed, is dictated by its x-coordinate and y-coordinate at that time, which we call x(t) and y(t) the coordinate functions, then it is safe to say that P(t)=(x(t), y(t)). This is what a parametric equation is. Instead of looking at how y is affected by x changing we are looking at how x and y are both affected by t changing. Although in the bug example t stood for time, that is not always the case. In general, t is called the parameter, and it can stand for anything as long as x and y are dependent upon the value of t. Sometimes we may want to think of t as an angle instead of as time, for instance. The real moral of the story to take away and that x(t) and y(t) are coordinate functions of some parameter t.
A very common use for parametric equations is for graphing curvy objects that are not easy to express in rectangular coordinates. In such cases the coordinate functions tend to take the form x(t)=cos(t) and y(t)=sin(t). Of course, constants can be added or multiplied into these functions in much the same way as in our first few investigations. Let's see how the graphs are affected by changing constants. First, we will explore when (x(t), y(t))=(acos(t), bsin(t)) for various a and b. Since cos(t) and sin(t) are periodic with a period of we will let t range from 0 to . The following graph is for a=1 and b=-3, -2, -1, 0
We don't see anything in the picture for the b=0 case but this is because our y(t)=0 which means the graph is a flat line segment that lies on top of the axis. So as b varies it appears that our ellipses is stretched in the verticle direction. Let's see what happens with positive values for b. In this graph we will look at b=0, 1, 2, 3.
Notice that the graphs look just like in the case before when we had negative values for b. Therefore, changing b does stretch the graph in a verticle direction but it does so without regard to the sign of b. This makes since because it is changing the range of y(t) from [-1, 1] to [-b, b].
I would predict that we will get similar results when we change the value of a, except with stretching in the horizontal direction since we are changing the x-coordinate function. When a=-3, -2, -1 and 0 we get
which is the same results that we get for a=0, 1, 2 and 3. Again, the range of x(t) is being changed from [-1, 1] to [-a, a]
When we change the parametric equation so that it is (x(t), y(t))=(cos(at), sin(bt)) the periods our our functions are changing while their ranges are staying [-1, 1]. Explore what happens as you change a and b when they appear in the forms (x(t), y(t))=(acost, bsint), (x(t), y(t))=(cos(at), sin(bt)) and (x(t), y(t))=(a+cost, b+sint) by going here. Conjecture why this is happening and try to validate your intuition with mathematical arguements. The point of this part of the investigation is just to introduce you to the concept of parametric equations and give you a feel for how they work. The next part is going to be solving a problem using parametric equations.
Imagine that you a bug taking a nap on a bicycle wheel. Suddenly, you are awoke when a ten-year old jumps on the bike and starts pedaling away! You are so shocked that you cannot fly off and instead hang onto the wheel for dear life. If your movement was tracked, what kind of curve would you trace out? Can I predict where you are assuming you were sleeping at the very bottom of the wheel when the bike took off, the path the bike is riding on is completely flat and I was given how much time you have traveled on the bike? That is what we are setting out to find. The name of the path that would be traced out is called a cycloid. Go here to see a dynamic picture of the situtation described as well as the locus of interest. In this case we are going to let our parameter t stand for the angle that is swept out (in radians) as the point moves along its path. In this way we will be able to figure out the position of the point at any given time. We will call the starting point of the bug (0, 0)
Imagine that before the bike took off it had recently gone through a puddle, so it was coated in mud. As the point of contact between the tire and the ground moves, the bike is going to leave a trail of mud from its starting point. In the picture that is represented by the dark blue line segment. The dark blue arc represents the part of the tire that has made contact with the road since the biker began to ride. Since the blue segment was formed by the blue arc, does it make sense that they should have equal lengths? Hopefully it does. Let's try to remember how we get arclength. The equation from trigonemtry is . In our case it would be s=rt. Therefore, the new point of contact between the tire and the road is a distance rt away from the starting point of contact. It still has height zero since it is on the road.
We know that the point of contact is directly below the center of the circle because we are on flat ground. That means the x coordiate of the center at any given time is equal to the x coordinate of the point of contact at that same time. By definition of the center of the circle we also know that is going to have a height of r. Again it is important here that the ground is flat. How would our picture be different if we were going up a hill? Down a hill?
In the picture you see a triangle the has been filled in. This triangle is a right triangle with radius r and an angle t. We want to try to fill in the rest of the triangle using the definitions of sine and cosine (do not just rely on rules of thumb learned about the unit circle). Let's fill out the triangle by itself, without the rest of the picture, so we don't get too cluttered in our work.
Now that we know the side lengths of our triangle we can go back and find the coordinates of our point of interest. Keep in mind, since our point is lower and to the left of our center, we are going to subtract the side lengths instead of adding them.
Great! We have found the coordinate functions for our parametric equation and can now say that
We can check that when we graph this function in Graphing Calculator we do indeed get a cycloid looking shape. The following is the graph when r=1/2.
Indeed, everything looks very good and we have solved our problem. Now try to solve it using rectangular coordinates so that you can fully appreciate the power of the parametric equation.
Parametric equations are very handy to understand and to be able to work with. Practice with them by taking equations that you know how to graph and figuring out a parametric representation for that graph (, , a straight line going through the points and , etc). Make different equations for x(t) and y(t) and play with the range of t. Understanding how parametric equations work can make a world of difference when learning higher level mathematics such as calculus and differential geometry.