Parametric equations of the form x = cos (at) and y = sin (bt), where a and b are constants , occur in electrical theory. The variables x and y usually represent voltages or currents at time t. The resulting curve is often difficult to sketch, but the graph can be represented on the screen of an oscilloscope when voltages or currents are imposed on the input terminals. These figures are called Lissajous curves; their complicated graphs can also be obtained using computer programs such as X-function used in this project. The investigation will deal with the types of these curves in which a and b are varied. The times will be restricted to t=0 to t = 2 pi.
First, let a = b = 1, so x = cos (t) and y = sin (t).
Here we have a circle with radius 1. Now let's try a = b= 2. That was exactly the same as the graph above. I tried several different integers here as well as fractions, but the graph was always a circle with radius 1. Recall that the coefficient of the whole term is the amplitude of the sine or cosine.
So now let's try a<b, say a=1 and b=2.
It looks like a bowtie! What if a=2 and b=3?
This looks like 2/3 of a bow tie, doesn't it? Let's try a=2 and
b=4 and then a=1 and b=4.
The a=2 and b=4 is exactly the same as a=1 with b=2. The same ratio between a and b seems to give the same graph. Now a=1 when b=4 gives us a double bow-tie or we could say 4 loops. Let's try a couple more: say a=1 when b=5 and a=1 when b=8.
It certainly appears as if there are the same number of bows as the ratio of b to a.
Now let's see what happens when a>b. We'll start with a=2 when b=1. On top of that we'll graph a=3 when b=2.
;With the first we get one loop that is open and with the second 3 closed loops. Notice that these loops are running horizontaly whereas the one's above were vertical. Let's try some more: say a = 5 when b =2 and then a=10 when b=3.
Notice that when the ratio is thirds the loops are not complete and are not symmetrical with the y axis, but when the ratio is halves the loops are complete and are symmetric with the y axis.
These are interesting curves, probably even more so for electricians.
Explorations:
1. Vary the coefficients of the sine and cosine as well as a and b.
2. Solve one of the above equations for x and y by using trigonometric identities and algebraic mainpulations.
3. Let x=sin(at) and y=cos(bt) and see what happens to the graph.