LISSAJOUS CURVE or BOWDITCH CURVE

 Those who see this movement in the red figure are, supposedly, "right-brain", those who see the oppose movement are "left-brain".... Why can't we see this movement in the red figure?

 Curve studied by Bowditch in 1815 and by Lissajous in 1857. Other names: Lissajous figure. Nathaniel Bowditch (1773-1838): American mathematician and sailor.  Jules Lissajous (1822-1880): French physicist.

 Reduced Cartesian parametrization:  ().

The Lissajous curves are the trajectories of a point the components of which have a sinusoidal movement.

The Lissajous curves of parameter n (ratio between the frequencies of the two sinusoidal movements) are the projections on the planes passing by the axis of the cylindric sine waves of parameter n:
as well as of the cylindric sine waves of parameter 1/n.
The curve whose reduced parametrization is in the header is indeed the projection on xOy of the cylindric sine wave of axis Oy and parameter n and the projection on xOy of the cylindric sine wave of axis Ox and parameter 1/.

If n is irrational, then the curve is dense in the rectangle .

If n is a rational number whose irreducible form is , then it is more convenient to use the following equations:
 Cartesian parametrization:  ,  . Algebraic curve of degree 2q if  when p is odd and  when p is even. Portion of an algebraic curve of degree q if  when p is odd, or if  when p is even. The number of double points is, in general, equal to  (p–1 groups of q points aligned on lines parallel to Ox, in blue opposite, and q–1 groups of p points aligned on lines parallel to Oy, in green opposite). In the case where the curve can be described in both directions, then there are  double points.

We get a portion of the plot of the n-th Chebyshev polynomial Tn when n is an even integer, and when n is an odd integer,  .
Here are some special cases, with a = b:

When n = 1, we get the ellipses:

When n = 2 (q = 2, p = 1), we get the besaces:

 : lemniscate of Gerono : portion of a parabola.
 projections of the cylindric sine wave of parameter 2 ( pancake curve) projections of the cylindric sine wave of parameter 1/2 ( Viviani's window)

When n = 3/2 (q = 3, p = 2):
 :  Sextic with Cartesian equation Portion of the divergent parabola with equation:.
 cylindric sine wave of parameter 3/2 cylindric sine wave of parameter 2/3

When n = 4/3, (q = 4, p = 3):

 Cartesian parametrization (curve on the right):  or  () Cartesian equation:  Polynomialquartic. See here a tied version of it.

 n = 5/3 n  = 5/4 n  = 6/5 n  = 8/5 n  = 9/8

 The Lissajous curves have the same topology as the curves of billiard balls in a rectangular billiard table. See this page.

 One can also imagine "Lissajous curves in polar coordinates", with polar parametrization:  ; opposite the case p = 3, q = 7,  (idea of Ch. de Rivière).

 This beautiful doormat does not follow exactly a Lissajous curve. Yet, if in the Lissajous curve , you follow the blue "bridges" opposite, you get this doormat. See an interpretation on this page.