Parametric Curves

(Assignment 10)

by

J. Matt Tumlin, Cara Haskins, and Robin Kirkham

 

Parametric curves in the plane   x = f(t) and y = g(t) are pairs of functions such that there are two continuous functions defined by an ordered pair (x, y).   These equations are usually called the parametric equations of a curve.  The extent of the curve depends on the range of t and the work with parametric equations while paying close attention to the range of t.  In many applications, think of x and y as they vary with respect to time t or the angle of rotation that some line makes from an initial location.

 

There are various technology that can be used to demonstrate these curves such as : TI-81, TI-82, TI-83, TI-85, TI-86, TI-89, Ohio state Grapher, xFunction, theorist, Graphing Calculator 3.2, and Derive.  This investigation is performed using the Graphing Calculator 3.2.

 

Graph

 

x = cos ( t )

y = sin ( t )          for    0   t    2p

 

 

 

As you observe the solution appears to be a circle with center at the origin and a radius of 1.

 

 

 

Further, let us observe the parametric equations

 

x = cos ( at )

y = sin ( bt )        for    0 t 2p

 

for various as and bs.

 

Let us observe some examples:

 

 

When a = b

 

        

 

        

 

The observation is that although the values of a and b are changed the circle remains about the origin with a radius of 1.

 

 

Lets look  at what happens when a = 2 and vary b in each graph such that b = 3, then b= 4, then b=5, and finally b= 6.

 

        

 

        

 

        

 

        

 

As we observe when a=2 ten the solution is a series of curves that look like loops.

 

The number of loops depends on what the value of b is set at. The number of loops = 1/2(b).

 

 

What  happens when b= 2 and we vary a as we have already done for b?

        

 

        

 

        

 

        

 

These graphs look very different; it appears that some of the oddity is when a is an odd number versa a is an even number.

 

When a is an odd number, there is always 2 local maximums and minimums for y and there is 1 maximum and minimum for x.

 

When a is an even number, there appears to be only one maximum and minimum for y and 1/2 of a maximums and minimums for x. When a=4, however, this is not true. Could there be a relationship change since at that point a=2b.

 

 

Let us observe what happens then when a=2b.

        

 

        

 

        

 

It seems true that when a=2b the graph is always in the above shape.

 

Conclusion:

This investigation shows a small sampling of what can be determined with the use of parametric curves. There are many interesting investigations that we could continue with yet this surely provides enough such that every new observation opens the doors to many other variations.

 

Simple using the basic curves and varying the a and b parameters to observe how the curves react provides us with much more that can be used and expanded on in the classroom.

 

 

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