Parametric Equations

Erin Mueller

Lissajous curves are a special class of parametric equations, which have the form  in which ÒaÓ and ÒbÓ can be any real number. LetÕs 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.

LetÕs look at different values for ÒaÓ and ÒbÓ. Below is the graph when ÒbÓ is changed to 2.  We have a bowtie.

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

As we can see, when ÒaÓ and ÒbÓ switch values, our graph rotates 90 degrees.

LetÕs explore other values of ÒaÓ and ÒbÓ.

When ÒaÓ and ÒbÓ are both odd, the graph is disconnected. Note that this only happens when ÒbothÓ 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 ÒbÓ. 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 ÒaÓ and ÒbÓ 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).