.. |newpage| raw:: latex \newpage .. |br| raw:: html
|newpage| General knots ============================================================== Trefoil A -------------------------------------------- See also: https://en.wikipedia.org/wiki/Trefoil_knot See also: https://katlas.org/wiki/T(3,2) An example in C\# .. code-block:: csharp 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); Some text |PathP_Trefoil_Knoten_Aa| `\quad` |PathO_Trefoil_Knoten_Aa| .. |PathP_Trefoil_Knoten_Aa| image:: ../_static/PathSurfaces/GeneralKnots/PathP_Trefoil_Knoten_Aa.3D.xml.jpg :width: 30 % .. |PathO_Trefoil_Knoten_Aa| image:: ../_static/PathSurfaces/GeneralKnots/PathO_Trefoil_Knoten_Aa.3D.xml.jpg :width: 30 % **Left figure**: Trefoil A. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Trefoil A. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. |newpage| Trefoil B -------------------------------------------- See also: https://en.wikipedia.org/wiki/Trefoil_knot See also: https://katlas.org/wiki/T(3,2) An example in C\# .. code-block:: csharp 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; Some text |PathP_Trefoil_Knoten_Ba| `\quad` |PathO_Trefoil_Knoten_Ba| .. |PathP_Trefoil_Knoten_Ba| image:: ../_static/PathSurfaces/GeneralKnots/PathP_Trefoil_Knoten_Ba.3D.xml.jpg :width: 30 % .. |PathO_Trefoil_Knoten_Ba| image:: ../_static/PathSurfaces/GeneralKnots/PathO_Trefoil_Knoten_Ba.3D.xml.jpg :width: 30 % **Left figure**: Trefoil B. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Trefoil B. Trefoil B. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 180^\circ`. |newpage| Square knot --------------------------------------------------- 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 An example in C\# .. code-block:: csharp 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; Some text |PathP_SquareKnot_P000T180_a| `\quad` |PathP_SquareKnot_P090T135_a| .. |PathP_SquareKnot_P000T180_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_SquareKnot_P000T180_a.3D.xml.jpg :width: 31 % .. |PathP_SquareKnot_P090T135_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_SquareKnot_P090T135_a.3D.xml.jpg :width: 40 % **Left figure**: Trefoil A. Perspective camera. Camera angles are `\theta=0^\circ` and `\phi = 180^\circ`. **Right figure**: Trefoil B. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 135^\circ`. Some text |PathO_SquareKnot_P000T180_a| `\quad` |PathO_SquareKnot_P090T135_a| .. |PathO_SquareKnot_P000T180_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_SquareKnot_P000T180_a.3D.xml.jpg :width: 30 % .. |PathO_SquareKnot_P090T135_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_SquareKnot_P090T135_a.3D.xml.jpg :width: 42 % **Left figure**: Trefoil A. Orthographic camera. Camera angles are `\theta=0^\circ` and `\phi = 180^\circ`. **Right figure**: Trefoil B. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 135^\circ`. |newpage| Granny knot -------------------------------------------------- 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 An example in C\# .. code-block:: csharp 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; Some text |PathP_GrannyKnot_P090T090_a| `\quad` |PathP_GrannyKnot_P090T000_a| .. |PathP_GrannyKnot_P090T090_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_GrannyKnot_P090T090_a.3D.xml.jpg :width: 30 % .. |PathP_GrannyKnot_P090T000_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_GrannyKnot_P090T000_a.3D.xml.jpg :width: 30 % **Left figure**: Trefoil A. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Trefoil B. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 0^\circ`. Some text |PathO_GrannyKnot_P090T090_a| `\quad` |PathO_GrannyKnot_P090T000_a| .. |PathO_GrannyKnot_P090T090_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_GrannyKnot_P090T090_a.3D.xml.jpg :width: 30 % .. |PathO_GrannyKnot_P090T000_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_GrannyKnot_P090T000_a.3D.xml.jpg :width: 30 % **Left figure**: Trefoil A. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Trefoil B. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 0^\circ`. |newpage| Cinquefoil Knot ------------------------------------------------ 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/ An example in C\# .. code-block:: csharp 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; Some text |PathP_CinquefoilKnot_P090T090_a| `\quad` |PathP_CinquefoilKnot_P000T180_a| .. |PathP_CinquefoilKnot_P090T090_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_CinquefoilKnot_P090T090_a.3D.xml.jpg :width: 30 % .. |PathP_CinquefoilKnot_P000T180_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_CinquefoilKnot_P000T180_a.3D.xml.jpg :width: 30 % **Left figure**: Cinquefoil Knot. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Cinquefoil Knot. Perspective camera. Camera angles are `\theta=180^\circ` and `\phi = 0^\circ`. Some text |PathO_CinquefoilKnot_P090T090_a| `\quad` |PathO_CinquefoilKnot_P000T180_a| .. |PathO_CinquefoilKnot_P090T090_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_CinquefoilKnot_P090T090_a.3D.xml.jpg :width: 30 % .. |PathO_CinquefoilKnot_P000T180_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_CinquefoilKnot_P000T180_a.3D.xml.jpg :width: 30 % **Left figure**: Cinquefoil Knot. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Cinquefoil Knot. Orthographic camera. Camera angles are `\theta=180^\circ` and `\phi = 0^\circ`. |newpage| 74 Knot ----------------------------------------------------- See also: https://en.wikipedia.org/wiki/74_knot This is a Lissajous knot. See also: https://katlas.org/wiki/7_4 An example in C\# .. code-block:: csharp 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; Some text |PathP_74Knot_P090T090_a| `\quad` |PathP_74Knot_P090T180_a| .. |PathP_74Knot_P090T090_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_74Knot_P090T090_a.3D.xml.jpg :width: 30 % .. |PathP_74Knot_P090T180_a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_74Knot_P090T180_a.3D.xml.jpg :width: 30 % **Left figure**: 74 Knot. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: 74 Knot. Perspective camera. Camera angles are `\theta=180^\circ` and `\phi = 90^\circ`. Some text |PathO_74Knot_P090T090_a| `\quad` |PathO_74Knot_P090T180_a| .. |PathO_74Knot_P090T090_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_74Knot_P090T090_a.3D.xml.jpg :width: 30 % .. |PathO_74Knot_P090T180_a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_74Knot_P090T180_a.3D.xml.jpg :width: 30 % **Left figure**: 74 Knot. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: 74 Knot. Orthographic camera. Camera angles are `\theta=180^\circ` and `\phi = 90^\circ`. |newpage| Figure-eight knot ------------------------------------------------------- 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 An example in C\# .. code-block:: csharp 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; Some text |PathP_AchterknotenP090T090a| `\quad` |PathP_Achterknoten_P090T180a| .. |PathP_AchterknotenP090T090a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_AchterknotenP090T090a.3D.xml.jpg :width: 30 % .. |PathP_Achterknoten_P090T180a| image:: ../_static/PathSurfaces/GeneralKnots/PathP_Achterknoten_P090T180a.3D.xml.jpg :width: 30 % **Left figure**: Figure-eight_knot. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Figure-eight_knot. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 180^\circ`. Some text Some text |PathO_AchterknotenP090T090a| `\quad` |PathO_Achterknoten_P090T180a| .. |PathO_AchterknotenP090T090a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_Achterknoten_P090T090a.3D.xml.jpg :width: 30 % .. |PathO_Achterknoten_P090T180a| image:: ../_static/PathSurfaces/GeneralKnots/PathO_Achterknoten_P090T000a.3D.xml.jpg :width: 30 % **Left figure**: Figure-eight_knot. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`. **Right figure**: Figure-eight_knot. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 180^\circ`.