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).