Lesson # 10
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.