Lissajous

Theorem. Given the parametric plane curve

where p, q are real numbers and m, n integers prime to each other.

Let
                                   .
Then the equation of the curve is
(3)
if m is even and ,
(4)
if m is even and ,
(5)
if m is odd and , and
(6)
if m is odd and .

Proof.
We will prove (3) and (4), the cases when m is even. When m is odd, (5) and (6) are obtained similarly.
Applying (2) with m even as well as the characteristic property of Chebyshev polynomials we obtain

Applying the formula for the cosine of the sum of two angles we get
                                                          (7)
This is a linear system in cos mnt and sin mnt, with determinant

If , applying Cramer’s rule we obtain the solution

           Substituting in the identity and observing that according to the formula for the cosine of the difference of two angles, the coefficient of is
     ,
we obtain (3) after multiplying the equation by
If the system (7) has a solution if
                          ,
and then

Adding the equations after multiplying the first by cos np and the second by sin np gives
,
or
,
which is (4).

Lissajous Figures and Chebyshev Polynomials
Julio Castiñeira Merino