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#

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D01a_Path_111_LKnot_P090T090.3D.xml (D01a-f).

The example below uses the following code 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;

D01a_Path_111_LKnot_P090T090.3D \(\quad\) D01b_Path_111_LKnot_P090T180.3D \(\quad\) D01c_Path_111_LKnot_P135T180.3D

D01d_Path_111_LKnot_P090T090.3D \(\quad\) D01e_Path_111_LKnot_P090T180.3D \(\quad\) D01f_Path_111_LKnot_P135T180.3D

Lissajous knot 1-2-1#

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D02a_Path_121_LKnot_P090T090.3D.xml (D02a-f).

The example below uses the following code 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;

D02a_Path_121_LKnot_P090T090.3D \(\quad\) D02b_Path_121_LKnot_P090T180.3D \(\quad\) D02c_Path_121_LKnot_P135T180.3D

D02d_Path_121_LKnot_P090T090.3D \(\quad\) D02e_Path_121_LKnot_P090T180.3D \(\quad\) D02f_Path_121_LKnot_P135T180.3D

Lissajous knot 1-5-3#

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D03a_Path_153_LKnot_P090T090.3D.xml (D03a-f).

The example below uses the following code 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;

D03a_Path_153_LKnot_P090T090.3D \(\quad\) D03b_Path_153_LKnot_P090T180.3D \(\quad\) D03c_Path_153_LKnot_P135T180.3D

D03d_Path_153_LKnot_P090T090.3D \(\quad\) D03e_Path_153_LKnot_P090T180.3D \(\quad\) D03f_Path_153_LKnot_P135T180.3D

Lissajous knot 3-5-2#

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D04a_Path_352_LKnot_P090T090.3D.xml (D04a-f).

The example below uses the following code 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;

D04a_Path_352_LKnot_P090T090.3D \(\quad\) D04b_Path_352_LKnot_P090T180.3D \(\quad\) D04c_Path_352_LKnot_P135T180.3D

D04d_Path_352_LKnot_P090T090.3D \(\quad\) D04e_Path_352_LKnot_P090T180.3D \(\quad\) D04f_Path_352_LKnot_P135T180.3D

Lissajous knot 3-5-7#

The XML code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the files D05a_Path_357_LKnot_P090T090.3D.xml (D05a-f).

The example below uses the following code 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;

D05a_Path_357_LKnot_P090T090.3D \(\quad\) D05b_Path_357_LKnot_P090T180.3D \(\quad\) D05c_Path_357_LKnot_P135T180.3D

D05d_Path_357_LKnot_P090T090.3D \(\quad\) D05e_Path_357_LKnot_P090T180.3D \(\quad\) D05f_Path_357_LKnot_P135T180.3D

Contents