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Polynomial knots
==============================================================

See also: https://home.adelphi.edu/~stemkoski/knotgallery/



Polynomial knot 3-1
----------------------------------------------------------------------



Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html


.. code-block:: csharp

    double t2 = t * t;
    double t3 = t2 * t;
    double t4 = t3 * t;
    double t5 = t4 * t;
    var x = (t3 - 3 * t) / 10;
    var y = (t4 - 4 * t2) / 10;
    var z = (t5 / 5 - 2 * t) / 10;


|PathP_Poly31Knot_a| `\quad` |PathO_Poly31Knot_a| 

.. |PathP_Poly31Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathP_Poly31Knot_a.3D.xml.jpg
   :width: 30 %

.. |PathO_Poly31Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathO_Poly31Knot_a.3D.xml.jpg
   :width: 30 %




**Left figure**: Polynomial knot 3-1. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.

**Right figure**: Polynomial knot 3-1. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.





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Polynomial knot 4-1
-----------------------------------------------------------


Figure-Eight (41) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html


.. code-block:: csharp

    double t2 = t * t;
    double t4 = t2 * t2;
    var x = (2.0 / 5.0) * t * (t2 - 7) * (t2 - 10) / 10;
    var y = (t4 - 13 * t2) / 10;
    var z = (1.0 / 10.0) * t * (t2 - 4) * (t2 - 9) * (t2 - 12) / 10;


|PathP_Poly41Knot_a| `\quad` |PathO_Poly41Knot_a| 

.. |PathP_Poly41Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathP_Poly41Knot_a.3D.xml.jpg
   :width: 30 %

.. |PathO_Poly41Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathO_Poly41Knot_a.3D.xml.jpg
   :width: 30 %




**Left figure**: Polynomial knot 4-1. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.

**Right figure**: Polynomial knot 4-1. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.




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Polynomial knot 5-1
----------------------------------------------------------


Cinquefoil (51) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html



.. code-block:: csharp

    double t2 = t * t;
    double t3 = t2 * t;
    double t4 = t3 * t;
    double t5 = t4 * t;
    double t6 = t5 * t;
    double t7 = t6 * t;
    var x = (1.0 / 5.0) * (t5 - 36 * t3 + 260 * t) / 10;
    var y = (1.0 / 2.0) * (t4 - 24 * t2) / 10;
    var z = (1.0 / 100.0) * (t7 - 31 * t5 + 164 * t3 - 560 * t) / 10;



|PathP_Poly51Knot_a| `\quad` |PathO_Poly51Knot_a| 

.. |PathP_Poly51Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathP_Poly51Knot_a.3D.xml.jpg
   :width: 30 %

.. |PathO_Poly51Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathO_Poly51Knot_a.3D.xml.jpg
   :width: 30 %




**Left figure**: Polynomial knot 5-1. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.

**Right figure**: Polynomial knot 5-1. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.





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Polynomial knot 6-2
---------------------------------------------------------


Six-Crossing (62) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html


.. code-block:: csharp

    double t2 = t * t;
    double t3 = t2 * t;
    double t4 = t3 * t;
    var x = (3.0 / 4.0) * t * (t2 - 4) * (t2 - 11) / 10;
    var y = (t4 - 12 * t2) / 10;
    var z = (1.0 / 200.0) * t * (t2 - 1) * (t2 - 3) * (t2 - 9) * (t2 - 10) * (t2 - 12) / 10;



|PathP_Poly62Knot_a| `\quad` |PathO_Poly62Knot_a| 

.. |PathP_Poly62Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathP_Poly62Knot_a.3D.xml.jpg
   :width: 30 %

.. |PathO_Poly62Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathO_Poly62Knot_a.3D.xml.jpg
   :width: 30 %




**Left figure**: Polynomial knot 6-2. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.

**Right figure**: Polynomial knot 6-2. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.





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Polynomial knot 7-4
--------------------------------------------------------


Seven-Crossing (74) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html



.. code-block:: csharp

    double t2 = t * t;
    double t3 = t2 * t;
    double t4 = t3 * t;
    double t5 = t4 * t;
    var x = (4.0 / 5.0) * t * (t2 - 6) * (t2 - 12) / 10;
    var y = t2 * (t2 - 7) * (t2 - 9) / 10;
    var z = (1.0 / 200.0) * t * (t2 - 0.2) * (t2 - 1) * (t2 - 5) * (t2 - 6.5) * (t2 - 9) * (t2 - 10) / 10;


|PathP_Poly74Knot_a| `\quad` |PathO_Poly74Knot_a| 

.. |PathP_Poly74Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathP_Poly74Knot_a.3D.xml.jpg
   :width: 30 %

.. |PathO_Poly74Knot_a| image:: ../_static/PathSurfaces/PolynomialKnots/PathO_Poly74Knot_a.3D.xml.jpg
   :width: 30 %




**Left figure**: Polynomial knot 7-4. Perspective camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.

**Right figure**: Polynomial knot 7-4. Orthographic camera. Camera angles are `\theta=90^\circ` and `\phi = 90^\circ`.