 This is the write-up for assignment #10.

Exploring Parametric Curves

by

Doug Westmoreland

This write-up explores a parametric curve in the plane. A parametric curve in the plane is a pair of functions, x=f(t) and y=g(t), where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent to which the curve will vary will depend on the range of the value of t.

Let's first consider a very basic set of parametric equations:

x=cos(t)

y=sin(t).

The graph of these two parametric equations for 0 < t < 6.28 is shown below. Now, let's consider the graph of:

x=acos(t)

y=bsin(t).

Let's take a look at several graphs where we let a and b vary in the parametric equations above.

First we will let the value of b remain 1 and we will let the value of a = -7(purple), -5(red), -3(blue), 2(green), 4(lt.blue), 6(yellow). We can see that the value of a will stretch the curve toward the right and left (while b remains a constant of 1).

Click here for a dynamic presentation where the value of a changes from -10 to 10.

Now, let's keep the value of a equal to 1 and vary the b value. We will let b =-7(purple), -5(red), -3(blue), 2(green), 4(lt.blue), 6(yellow). We can see that the value of b will stretch the curve up and down (while a remains a constant of 1).Click here for a dynamic presentation where the value of b changes from -10 to 10.

Now, let's take a look at several graphs where we let a and b both vary at the same time.

This is the graph of x=2cos(t) and y=6sin(t): This is the graph of x=5cos(t) and y=2sin(t): This is the graph of x=-7cos(t) and y=3sin(t): This is the graph of x=-2cos(t) and y=-sin(t): *******************************************

After exploring the parametric curves of x=acos(t) and y=bsin(t), it looks as though the a and b values will determine the intercepts on the x and y axis. The a value will determine the x-intercept and the b value will determine the y-intercept. Also, when a<b, the graph is elliptical with the major axis on the y-axis. When a>b, the graph is elliptical with the major axis on the x-axis. And when a=b, the graph is a circle with center at the origin.

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Now, let's explore various values of a and b in the parametric equation:

x=cos(at)

y=sin(bt).

We will first let b value be 1 and we will vary the a value.

Here is the graph of x=cos (-5t) and y=sin(t): This is also the graph of x=cos(5t) and y=sin(t).

Here is the graph of x=cos (-3t) and y=sin(t): Also, this is the graph of x=cos(3t) and y= sin(t).

Here is the graph of x=cos(-1t) and y=sin(t): Once again, this is also the graph when a=1 in x=cos(at) and y=sin(t).

Here is the graph of x=cos(-.5t) and y=sin(t): This is also the graph for x=cos (.5t) and y=sin(t).

Here are the graphs of x=cos(at) and y=sin(t) for a=-.4 (purple), -.3 (red), -.2 (blue), -.1 (green), 0 (lt.blue). If we let the values of a = .4, .3, .2, .1, we will get the same graphs as above.

** Click here for a dynamic presentation where the value of a changes from -5 to 5.

Now, let the a value be 1 and we will vary the b value.

Here is the graph of x=cos (t) and y=sin(-5t): This is also the graph of x=cos(t) and y=sin(5t). (This is the reflection of the above graph over the x-axis.)

Here is the graph of x=cos (t) and y=sin(-3t): Also, this is the graph of x=cos(t) and y= sin(3t). (This is the reflection of the above graph over the x-axis.)

Here is the graph of x=cos(t) and y=sin(-1t): Once again, this is also the graph when b=1 in x=cos(t) and y=sin(bt). (This is the reflection of the above graph over the x-axis.)

Here is the graph of x=cos(t) and y=sin(-.5t): If we looked at the graph for x=cos (t) and y=sin(.5t), we would reflect it over the x-axis like: Here are the graphs of x=cos(t) and y=sin(bt) for b=-.4 (purple), -.3 (red), -.2 (blue), -.1 (green), 0 (lt.blue). If we let the values of b = .4, .3, .2, .1, we will get this graph: Which is once again a reflection over the x-axis.

** Click here for a dynamic presentation where the value of b changes from -5 to 5.

If we look at varying the a and b values at the same time, we could generate an infinite number of graphs. One thing that I observed is that when the a and b values are both the same in x=cos(at) and y=sin(bt) (a>1) then a graph of a circle is always formed. When 0<a<1 , then the graph is a fraction of the circle. For example, if x=cos(.25t) and y=sin(.25t), then 1/4 of the total circle with radius 1 and center at the origin is graphed. Return