where 0 < b < π/2.
When the constant a is rational, the curve is algebraic and
closed.
If a is irrational, the curve fills the area [1,1] x [1,1].
It is easy to see that the curves for 1/a and a are equal in form. This means that we can
confine ourselves to the case a < 1.
JulesAntoine Lissajous (18221880) discovered these elegant curves (in 1857) while doing
his sound experiments.
But it is said that the American Nathaniel Bowditch (17731838)
found the curves already in 1815. After him the curve bears the name of
Bowditch curve.
Another name that I found is the play curve of Alice^{1}.
The curves are constructed as a combination of two perpendicular harmonic
oscillations. Patterns occur as a result of differences in frequency ratio (a) and phase (b).
At high school we used the oscilloscope to make the curve visible (nowadays a computer
would do), by connecting different harmonic signals to the x and yaxis
entrance. The curves have applications in physics, astronomy and other sciences.
Each Lissajous curve can be described with an algebraic equation.
Write a as the smallest integers m, n for which a = m/n. Then the degree of the equation obeys the following rules:
 degree = m
for b = 0 and m is odd
for b = π/2 and m is even
 degree = 2m
for b ∈ ]0,π/2] and m is odd
for b ∈ [0,π/2[ and m is even
Some examples of Lissajous curves are the following:
a 
b 
name of the curve 
1 
0 
straight line 
1 
π/2 
circle 
1 
<> 0, π/2 
ellipse 
2 
π/2 
parabola

2 
<> π/2 
besace: its
parameter a is equal to the tangent of the Lissajous curve's parameter b. 
3 
0 
the cubic y = 2x^{3}  x (see the picture below) 
3 
π/2 
the sextic y^{2}
= (1  x^{2})(14x^{2})^{2} (see the picture below) 
In the situation that the parameter a is an integer, three forms of the curve can be distinguished:
 the curve is symmetric with respect to the x and yaxis
When a is even: b = 0. When a is odd: b = π/2.
 the curve is asymmetric
For this form (and the one before, as a matter of fact) the number of compartments is
equal to the value of a.
 the curve has no compartments, the two parts of the curve come
together
When a is even:
b = π/2. When a is odd: b = 0.
There is a relation with the Tchebyscheff
polynomial: the Lissajous curve (a, b) corresponds with Tchebyscheff
function T_{a} for a integer and b=π/2 (for even a) or b=0 (for odd a).
The Lissajous curve can be extended to the
notes
1) In French: joue courbe d'Alice.
