Lissajous Curves

Lissajous Curves

Lissajous Curves were explored by French Physicist Jules Lissajous in the 1850s.  The picture above (Wikimedia Commons) shows him investigating Lissajous curves through a telescope.

Lissajous curves include those which can be written in the form:

This parametric form allows us to represent complicated curves which are difficult to write in terms of x and y only. 

A simple case:

Let’s see a simple case, when n = 1 and a = b:

We can there write this as:

And so if we square both equations we get:

And now we can add both equations:

We use the identity that sin squared theta + cos squared theta = 1.  This gives us the equation of a circle centred (0,0) with radius a.

More interesting shapes!

We can then fix some of the variables and explore on Desmos what shapes we can create.  Desmos can handle parametric equations by using the following notation:

Let’s see what happens when we fix a = 3 and n = 4 and then vary the value of b:

b = 2

We can see that as b varies we get a vertical stretch and the value of b dictates the maximum and minimum on the y axis.  By the same idea the value of a will dictate the horizontal stretch and the maximum and minimum values in the x direction.

Next let’s fix a = 3 and b =3 and see what happens when we vary n.

n = 0.4

n = 1

n = 2

n = 2.9