Lissajous

Lissajous Figures and Chebyshev Polynomials
Julio Castiñeira Merino
jcastine@boj.pntic.mec.es
published in
The College Mathematics Journal
Vol 34, No 2, March 2003
Copyright the Mathematical Association
of America 2003. All rigths reserved

To my wife Julia and my children Isabel, Lucía and Rodrigo

          The aim of this paper is to obtain implicit equations for Lissajous figures in the form using some elementary properties of Chebyshev polynomials.

          A Lissajous figure is the trajectory of a moving point whose rectangular coordinates are simple harmonic motions. The equation of a simple harmonic motion is where t is the time and the constants a, ω, and   are the amplitude, the frequency, and the phase respectively. Parametric equations for Lissajous figures thus are


          The constants a and b determine the size of the curve while its shape depends on the ratio of the frequencies. If the frequencies are equal, the curve is either an ellipse or a line segment, the latter occurring if the difference of the phases is an multiple of π. This property can be used to study an unknown signal. If we apply the unknown signal to the vertical axis of an oscilloscope and then vary the horizontal frequency, when the oscilloscope shows an ellipse the signal’s frequency has been determined.

          Lissajous figures were actually discovered by the American astronomer and mathematician Nathaniel Bowditch in 1815 when he was studying the movement of a compound pendulum. Bowditch (1773-1838) was a self-taught scientist, a captain of a merchant ship, president of an insurance company, actuary for the Massachusetts Hospital Insurance Company of Boston, and president of the American Academy of Arts and Sciences. Author of a number of scientific works, he is remembered mostly for his book The New American Practical Navigator, which was adopted by the U.S. Department of the Navy. Jules Antoine Lissajous (1822-1880) was a French physicist who extensively studied the curves that bear his name independently of Bowditch during his research on optics [2].

         Among the different ways of writing the parametric equations for Lissajous figures, or curves (a more natural term), we choose the following, where m and n are integers, prime to each other, p and q are real numbers, and
(1)

          Obviously any figure whose ratio of frequencies is rational can be expressed in this way by making a linear change of variable. Expressing the abscissa as a function of the sine and the ordinate as a function of the cosine is done for a purpose. It is like the parametrization of the circle and it allows us to express the equation of a Lissajous figure in a simple way.

Lissajous Figures and Chebyshev Polynomials
Julio Castiñeira Merino