Lissajous curves

We consider the ordinary differential equationMathworldPlanetmath

dydx=±kb2-y2a2-x2, (1)


(dydx)2=k2b2-y2a2-x2, (2)

where a, b and k are positive .  It’s evident that the equations

x=±a,y=±b (3)

give singular solutions of (1).  The lines (3) divide the xy-plane into nine parts, each one of which contains a family of integral curves of (1).

Denote by R the rectangleMathworldPlanetmath|x|a,|y|b.  After separation of variablesMathworldPlanetmath, we can write in R the differential equation as


which leads to the general integral

arcsinyb=±karcsinxa+C (4)

and therefore

y=±bsin(karcsinxa+C). (5)

The equation (5) with the plus sign represents a family of smooth arcs once the rectangle R.  Every single arc two opposite sides of R which are tangent lines of the arc.  The situation is analogical in the case of the minus sign.  One arc of both kinds passes through every interior point of R.

For better examining the curves (5) one may take


for a parametre.  Then,  x=asint,  and (5) may be replaced by

{x=asint,y=bsin(kt+C). (6)

Letting t to change freely, for any given value of C the equations (6) represent a continuous curve formed by an arc from the first family and another arc from the second family.  All such curves (6) are integral curves of the equation (2) and are called Lissajous curvesMathworldPlanetmath.

It may be shown that for any rational value of k, (6) is a smooth closed curveMathworldPlanetmath, except when the curve comes to a vertex of the rectangle R.  If the value of k is irrational, then (6) is never a closed curve, and any such curve fills the whole rectangle in the sense that it comes arbitrarily to every point of R.  In the former case, all integral curves of (2) are algebraicMathworldPlanetmath (

Some figures in


  • 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title Lissajous curves
Canonical name LissajousCurves
Date of creation 2013-03-22 19:13:33
Last modified on 2013-03-22 19:13:33
Owner rrogers (21140)
Last modified by rrogers (21140)
Numerical id 13
Author rrogers (21140)
Entry type Example
Classification msc 34C25
Classification msc 34A05
Related topic SawBladeFunction
Related topic SequenceAccumulatingEverywhereIn11