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?

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/n .
 

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 T_n 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

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.

See also the 3D Lissajous curves, and the basins.
 
 
 

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© Robert FERRÉOL 2017