Lissajous knots

Lissajous knots#

See also: https://en.wikipedia.org/wiki/Lissajous_knot

See also: https://mathcurve.com/courbes3d.gb/lissajous3d/noeudlissajous.shtml

See also: https://knotplot.com/manual/LissajousParams.html

The Lissajous knots are the knots associated to the 3D Lissajous curves when they are closed and without double point.

As is proved in the above article by Jones and Przytycki, they also are the knots associated to the trajectories of a ball (not subject to gravity) in a parallelepipedic billiard, or even a cubic one (imagine a glass box).

It can be proved that for all values of p, q, r, there exist values of j and y such that the knot is trivial, and that certain knots such as the trefoil knot are not Lissajous knots.

See also: https://mathcurve.com/courbes3d.gb/lissajous3d/lissajous3d.shtml

The 3D Lissajous curves are the trajectories of a point in space the rectangular components of which have a sinusoidal motion. The projections on the 3 coordinate planes are the classic 2D Lissajous curves.

For n = 1 or n = m, we get a cylindrical sine wave. We get a closed curve if and only if n and m are rational.

When the curve does not have double points, nor a cusp, it forms a knot in space, called Lissajous knot, equivalent to a cubic billiard knot.

Lissajous knot 1-1-1#

An example in C#

double pi = Math.PI;
double a1, k1, l1;
double a2, k2, l2;
double a3, k3, l3;
a1 = 100; k1 = 1; l1 = 0;
a2 = 100; k2 = 1; l2 = pi / 2;
a3 = 100; k3 = 1; l3 = pi / 2;
var x = (a1 * Math.Cos(k1 * t + l1)) / 50;
var y = (a3 * Math.Cos(k3 * t + l3)) / 50;
var z = (a2 * Math.Cos(k2 * t + l2)) / 50;

PathP_111_LKnot_P090T090a \(\quad\) PathP_111_LKnot_P090T180a \(\quad\) PathP_111_LKnot_P135T180a

Left figure: Lissajous knot 1-1-1. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 1-1-1. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 1-1-1. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Some text

PathO_111_LKnot_P090T090a \(\quad\) PathO_111_LKnot_P090T180a \(\quad\) PathO_111_LKnot_P135T180a

Left figure: Lissajous knot 1-1-1. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 1-1-1. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 1-1-1. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Lissajous knot 1-2-1#

An example in C#

double pi = Math.PI;
double a1, k1, l1;
double a2, k2, l2;
double a3, k3, l3;
a1 = 100; k1 = 1; l1 = 0;
a2 = 100; k2 = 2; l2 = pi / 2;
a3 = 100; k3 = 1; l3 = pi / 2;
var x = (a1 * Math.Cos(k1 * t + l1)) / 50;
var y = (a3 * Math.Cos(k3 * t + l3)) / 50;
var z = (a2 * Math.Cos(k2 * t + l2)) / 50;

PathP_121_LKnot_P090T090a \(\quad\) PathP_121_LKnot_P090T180a \(\quad\) PathP_121_LKnot_P135T180a

Left figure: Lissajous knot 1-2-1. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 1-2-1. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 1-2-1. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Some text

PathO_121_LKnot_P090T090a \(\quad\) PathO_121_LKnot_P090T180a \(\quad\) PathO_121_LKnot_P135T180a

Left figure: Lissajous knot 1-2-1. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 1-2-1. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 1-2-1. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Lissajous knot 1-5-3#

An example in C#

double pi = Math.PI;
double a1, k1, l1;
double a2, k2, l2;
double a3, k3, l3;
a1 = 100; k1 = 1; l1 = 0;
a2 = 100; k2 = 5; l2 = pi / 2;
a3 = 100; k3 = 3; l3 = pi / 2;
var x = (a1 * Math.Cos(k1 * t + l1)) / 50;
var y = (a3 * Math.Cos(k3 * t + l3)) / 50;
var z = (a2 * Math.Cos(k2 * t + l2)) / 50;

PathP_153_LKnot_P090T090a \(\quad\) PathP_153_LKnot_P090T180a \(\quad\) PathP_153_LKnot_P135T180a

Left figure: Lissajous knot 1-5-3. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 1-5-3. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 1-5-3. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Some text

PathO_153_LKnot_P090T090a \(\quad\) PathO_153_LKnot_P090T180a \(\quad\) PathO_153_LKnot_P135T180a

Left figure: Lissajous knot 1-5-3. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 1-5-3. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 1-5-3. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Lissajous knot 3-5-2#

An example in C#

double pi = Math.PI;
double a1, k1, l1;
double a2, k2, l2;
double a3, k3, l3;
a1 = 100; k1 = 3; l1 = 0;
a2 = 100; k2 = 5; l2 = pi / 2;
a3 = 100; k3 = 2; l3 = pi / 2;
var x = (a1 * Math.Cos(k1 * t + l1)) / 50;
var y = (a3 * Math.Cos(k3 * t + l3)) / 50;
var z = (a2 * Math.Cos(k2 * t + l2)) / 50;

PathP_352_LKnot_P090T090a \(\quad\) PathP_352_LKnot_P090T180a \(\quad\) PathP_352_LKnot_P135T180a

Left figure: Lissajous knot 3-5-2. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 3-5-2. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 3-5-2. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Some text

PathO_352_LKnot_P090T090a \(\quad\) PathO_352_LKnot_P090T180a \(\quad\) PathO_352_LKnot_P135T180a

Left figure: Lissajous knot 3-5-2. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 3-5-2. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 3-5-2. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Lissajous knot 3-5-7#

An example in C#

double pi = Math.PI;
double a1, k1, l1;
double a2, k2, l2;
double a3, k3, l3;
a1 = 100; k1 = 3; l1 = 7;
a2 = 100; k2 = 5; l2 = 5;
a3 = 100; k3 = 7; l3 = 3;
var x = (a1 * Math.Cos(k1 * t + l1)) / 50;
var y = (a3 * Math.Cos(k3 * t + l3)) / 50;
var z = (a2 * Math.Cos(k2 * t + l2)) / 50;

PathP_357_LKnot_P090T090a \(\quad\) PathP_357_LKnot_P090T180a \(\quad\) PathP_357_LKnot_P135T180a

Left figure: Lissajous knot 3-5-7. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 3-5-7. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 3-5-7. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

Some text

PathO_357_LKnot_P090T090a \(\quad\) PathO_357_LKnot_P090T180a \(\quad\) PathO_357_LKnot_P135T180a

Left figure: Lissajous knot 3-5-7. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Middle figure: Lissajous knot 3-5-7. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Right figure: Lissajous knot 3-5-7. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 135^\circ\).

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