General knots

General knots#

Trefoil A#

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_Trefoil_Knoten_A.3D.xml (D01a-d).

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

See also: https://katlas.org/wiki/T(3,2)

The example below uses the following code in C#

var x = (2 + Math.Cos(3 * t)) * Math.Cos(2 * t);
var y = (2 + Math.Cos(3 * t)) * Math.Sin(2 * t);
var z = Math.Sin(3 * t);

D01a_Path_Trefoil_Knoten_A.3D \(\quad\) D01b_Path_Trefoil_Knoten_A.3D

D01c_Path_Trefoil_Knoten_A.3D \(\quad\) D01d_Path_Trefoil_Knoten_A.3D

Trefoil B#

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_Trefoil_Knoten_B.3D.xml (D02a-d).

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

See also: https://katlas.org/wiki/T(3,2)

The example below uses the following code in C#

var x = (-10 * Math.Cos(t) - 2 * Math.Cos(5 * t) + 15 * Math.Sin(2 * t)) / 10;
var y = (-15 * Math.Cos(2 * t) + 10 * Math.Sin(t) - 2 * Math.Sin(5 * t)) / 10;
var z = (10 * Math.Cos(3 * t)) / 10;

D02a_Path_Trefoil_Knoten_B.3D \(\quad\) D02b_Path_Trefoil_Knoten_B.3D

D02c_Path_Trefoil_Knoten_B.3D \(\quad\) D02d_Path_Trefoil_Knoten_B.3D

Square knot#

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_SquareKnotP_P90T90.3D.xml (D03a-f).

See also: https://en.wikipedia.org/wiki/Square_knot_(mathematics)

See also: https://mathworld.wolfram.com/SquareKnot.html

See also: https://mathcurve.com/courbes3d.gb/plat.vache/plat_vache.shtml

The example below uses the following code in C#

var x = (-22 * Math.Cos(t) - 128 * Math.Sin(t) - 44 * Math.Cos(3 * t) - 78 * Math.Sin(3 * t)) / 10;
var y = (11 * Math.Cos(t) - 43 * Math.Cos(3 * t) + 34 * Math.Cos(5 * t) - 39 * Math.Sin(5 * t)) / 10;
var z = (70 * Math.Cos(3 * t) - 40 * Math.Sin(3 * t) + 8 * Math.Cos(5 * t) - 9 * Math.Sin(5 * t)) / 10;

D03a_Path_SquareKnotP_P90T90.3D \(\quad\) D03b_Path_SquareKnot.3D \(\quad\) D03c_Path_SquareKnotO_P90T90.3D

D03d_Path_SquareKnotP_P0T180.3D \(\quad\) D03e_Path_SquareKnot.3D \(\quad\) D03f_Path_SquareKnot.3D

Granny knot#

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_GrannyKnot.3D.xml (D04a-f).

See also: https://en.wikipedia.org/wiki/Granny_knot_(mathematics)

See also: https://mathworld.wolfram.com/GrannyKnot.html

See also: https://mathcurve.com/courbes3d.gb/plat.vache/plat_vache.shtml

The example below uses the following code in C#

var x = (-22 * Math.Cos(t) - 128 * Math.Sin(t) - 44 * Math.Cos(3 * t) - 78 * Math.Sin(3 * t)) / 10;
var y = (-10 * Math.Cos(2 * t) - 27 * Math.Sin(2 * t) + 38 * Math.Cos(4 * t) + 46 * Math.Sin(4 * t)) / 10;
var z = (70 * Math.Cos(3 * t) - 40 * Math.Sin(3 * t)) / 10;

D04a_Path_GrannyKnot.3D \(\quad\) D04b_Path_GrannyKnot.3D \(\quad\) D04c_Path_GrannyKnot.3D

D04d_Path_GrannyKnot.3D \(\quad\) D04e_Path_GrannyKnot.3D \(\quad\) D04f_Path_GrannyKnot.3D

Cinquefoil Knot#

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_CinquefoilKnot.3D.xml (D05a-d).

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

See also: https://katlas.org/wiki/T(5,2)

Cinquefoil Knot: https://home.adelphi.edu/~stemkoski/knotgallery/

The example below uses the following code in C#

var x = Math.Cos(2 * t) * (3 + Math.Cos(5 * t)) / 10;
var y = Math.Sin(2 * t) * (3 + Math.Cos(5 * t)) / 10;
var z = Math.Sin(5 * t) / 10;

D05a_Path_CinquefoilKnot.3D \(\quad\) D05b_Path_CinquefoilKnot.3D

D01c_Path_Trefoil_Knoten_A.3D \(\quad\) D05d_Path_CinquefoilKnot.3D

74 Knot#

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 D06a_Path_74Knot.3D.xml (D06a-f).

See also: https://en.wikipedia.org/wiki/74_knot

This is a Lissajous knot.

See also: https://katlas.org/wiki/7_4

The example below uses the following code in C#

var x = Math.Cos(2 * t + 0.22) / 10;
var y = Math.Cos(3 * t + 1.10) / 10;
var z = Math.Cos(7 * t) / 10;

D06a_Path_74Knot.3D \(\quad\) D06b_Path_74Knot.3D \(\quad\) D06c_Path_74Knot.3D

D06d_Path_74Knot.3D \(\quad\) D06e_Path_74Knot.3D \(\quad\) D06f_Path_74Knot.3D

Figure-eight knot#

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 D07a_Path_FigureEightKnot.3D.xml (D07a-f).

See also: https://mathcurve.com/courbes3d.gb/noeuds/noeudenhuit.shtml

See also: https://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)

See also: https://katlas.org/wiki/4_1

The example below uses the following code in C#

var x = ((2 + Math.Cos(2 * t)) * Math.Cos(3 * t)) / 10;
var y = ((2 + Math.Cos(2 * t)) * Math.Sin(3 * t)) / 10;
var z = (Math.Sin(4 * t)) / 10;

D07a_Path_FigureEightKnot.3D \(\quad\) D07b_Path_FigureEightKnot.3D \(\quad\) D07c_Path_FigureEightKnot.3D

D07d_Path_FigureEightKnot.3D \(\quad\) D07e_Path_FigureEightKnot.3D \(\quad\) D07f_Path_FigureEightKnot.3D

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