Parametric Equations

Erin Mueller

Lissajous curves are a special class of parametric equations, which have the form in which “a” and “b” can be any real number. Let’s look at the graphs of these special functions.

Below is the graph when a=1 and b=1.

As we can see, there is simply a line segment.

Let’s look at different values for “a” and “b”. Below is the graph when “b” is changed to 2. We have a bowtie.

Now let’s see what happens when we have a natural number. We can make a=2 and b=1.

As we can see, when “a” and “b” switch values, our graph rotates 90 degrees.

Let’s explore other values of “a” and “b”.

When “a” and “b” are both odd, the graph is disconnected. Note that this only happens when “both” of the variables are odd.

If either one of the variables is even, our graph is still connected.

Now, if we notice the number of points that correspond to the maximum and minimum y-values are based on the value of “b”. In the above graph, there are 5 x-values that correspond to the maximum and minimum y-values for the function. The coordinates for these points are (-4,3), (-2.5,3), (0,3), (2.5,3), and (4,3) (Note: these are only the positive points). Notice that the maximum and minimum y-values (3 and -3) correspond to the function y=3sin(t). There are 4 y-values that correspond to the maximum and minimum x-values for the function. The coordinates for these points are (4,3), (4,2.5), (4, -2.5), and (4, -3). It seems the x and y-coordinates have simply switched places. This is due to the symmetry properties of this type of function. However, this symmetry does not exist when “a” and “b” are both odd. Again, for these coordinate points; notice the x-values are all positive 4. There are also four more coordinates not listed but the only difference is the x-coordinate would be -4. This corresponds to the function x=4sin(t).

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