Department of Mathematics
J.Wilson, EMAT 6680

Lesson # 10
Parametric Equations
by Jan White

An exploration of the graph: x=cos(at)
                                           y=sin(bt),  for various values of a and b.

When a = b, this will draw a graph of a circle with radius of 1.

                                           
When both a and b are negative the same graph appears.

Next, an examination of the parametric equations : a cos t
                                                                             b sin t, for various values.

First let's look at when a=2 and b=1, and when a=1 and b=2.

As you can see, the circle has been elongated along the x-axis when a=2 and along the y-axis when b=2.    

An examination of the graphs of parametric equations:  x=cos(at)
                                                                                   y=sin(bt), for various values when a does not =b.

Both of these graphs are symmetric to the x-axis.  The second graph is commonly refered to as a Lissajous Curve.
Will the graph change if a and b are negative?  No.

 

What will happen when "sin" and "cos" are swithched?  They are now symmetric with respect to the y-axis.

Finally, what happens to the graphs of the parametric equations when a and b change and the coefficients of "cos" and "sin" change?

The same Lissajous Curve appears but has been stretched along the x=axis by 2 and the y-axis by 3 just as the circles before were stretched.

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