.. |newpage| raw:: latex \newpage .. |br| raw:: html
|newpage| Nonorientable (one-sided) Surfaces ======================================== Moebius Strip ------------------------------------------ See also: http://en.wikipedia.org/wiki/M%C3%B6bius_strip See also: https://mathworld.wolfram.com/MoebiusStrip.html .. code-block:: csharp x = (1 + (v / 2) * Math.Cos(u / 2)) * Math.Cos(u); z = (1 + (v / 2) * Math.Cos(u / 2)) * Math.Sin(u); y = (v / 2) * Math.Sin(u / 2); |TestMoebius_a| `\quad` |TestMoebius_b| .. |TestMoebius_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestMoebius_a.3D.xml.jpg :width: 30 % .. |TestMoebius_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestMoebius_a.3D.xml.jpg :width: 30 % **Left figure**: Moebius Strip (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Moebius Strip (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Cross-Cap Surface ------------------------------ See also: https://mathworld.wolfram.com/Cross-Cap.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. .. code-block:: csharp double su = Math.Sin(u); double sv = Math.Sin(v); double s2v = Math.Sin(2*v); double cu = Math.Cos(u); double cv = Math.Cos(v); x = 0.5 * cu * s2v; z = 0.5 * su * s2v; y = 0.5 * (cv*cv - cu*cu * sv*sv); |TestCrossCap_a| `\quad` |TestCrossCap_b| .. |TestCrossCap_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestCrossCap_a.3D.xml.jpg :width: 30 % .. |TestCrossCap_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestCrossCap_a.3D.xml.jpg :width: 30 % **Left figure**: Cross-Cap Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Cross-Cap Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Pseudo Cross-Cap Surface -------------------------------------------- See also: https://mathworld.wolfram.com/Pseudocrosscap.html See also: http://www.3d-meier.de/tut3/Seite51.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. .. code-block:: csharp double sv = Math.Sin(v); double s2v = Math.Sin(2*v); x = (1 - u*u) * sv; y = (1 - u*u) * s2v; z = u; |TestPseudoCrossCap_a| `\quad` |TestPseudoCrossCap_b| .. |TestPseudoCrossCap_a| image:: ../_static/ParametricSurfaces/Removal/TestPseudoCrossCap_a.3D.xml.jpg :width: 30 % .. |TestPseudoCrossCap_b| image:: ../_static/ParametricSurfaces/Removal/TestPseudoCrossCap_a.3D.xml.jpg :width: 30 % **Left figure**: Pseudo Cross-Cap Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Pseudo Cross-Cap Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Roman Surface (or Steiner Surface) ----------------------------------------------- See also: https://mathworld.wolfram.com/RomanSurface.html See also: https://en.wikipedia.org/wiki/Roman_surface See also: http://paulbourke.net/geometry/steiner/ .. code-block:: csharp double r2 = 1; double su = Math.Sin(u); double sv = Math.Sin(v); double cu = Math.Cos(u); double cv = Math.Cos(v); x = r2 * cu * su * sv; y = r2 * cu * su * cv; z = r2 * cu*cu * sv * cv; See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |TestRoman_a| `\quad` |TestRoman_b| .. |TestRoman_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestRoman_a.3D.xml.jpg :width: 30 % .. |TestRoman_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestRoman_a.3D.xml.jpg :width: 30 % **Left figure**: Roman Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Roman Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Klein bagel ----------------------------- See also: https://en.wikipedia.org/wiki/Klein_bottle#The_figure_8_immersion See also: https://mathworld.wolfram.com/KleinBottle.html .. code-block:: csharp double r = 2.1; x = (r + Math.Cos(u / 2) * Math.Sin(v) - Math.Sin(u / 2) * Math.Sin(2 * v)) * Math.Cos(u); z = (r + Math.Cos(u / 2) * Math.Sin(v) - Math.Sin(u / 2) * Math.Sin(2 * v)) * Math.Sin(u); y = Math.Sin(u / 2) * Math.Sin(v) + Math.Cos(u / 2) * Math.Sin(2 * v); This is the 'bagel' form of a Klein bottle, a 4 dimensional object with a single surface (lacking 'inside' or 'outside'), projected into 3-space as a self-intersecting solid. |TestKleinBagel_a| `\quad` |TestKleinBagel_b| .. |TestKleinBagel_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBagel_a.3D.xml.jpg :width: 30 % .. |TestKleinBagel_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBagel_a.3D.xml.jpg :width: 30 % **Left figure**: Klein bagel (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Klein bagel (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Klein bottle, version 3 --------------------------------------- See also: http://www.mapleprimes.com/maplesoftblog/95570-Klein-Bottle-Plot See also: http://www.chebfun.org/examples/geom/ParametricSurfaces.html See also: https://mathworld.wolfram.com/KleinBottle.html .. code-block:: csharp double a = Math.Cos(u); double b = Math.Sin(u); double c = Math.Cos(v); double a2 = a * a; double a4 = a2 * a2; x = -(2.0 / 15.0) * a * (3 * c + b * (-30 + a4 * (90 - 60 * a2) + 5 * a * c)); z = -(1.0 / 15.0) * b * b * (c * b * (3 - 48 * a4 + 5 * a * b * (1 - 16 * a4)) - 60); y = -(2.0 / 15.0) * (3 + 5 * a * b) * Math.Sin(v); |TestKleinBottle3_a| `\quad` |TestKleinBottle3_b| .. |TestKleinBottle3_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBottle3_a.3D.xml.jpg :width: 30 % .. |TestKleinBottle3_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBottle3_a.3D.xml.jpg :width: 30 % **Left figure**: Klein bottle, version 3 (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Klein bottle, version 3 (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Klein bottle, version 2 ----------------------------------- See also: https://mathworld.wolfram.com/KleinBottle.html The following parametrization is from the English Wikipedia article: https://en.wikipedia.org/wiki/Klein_bottle#Bottle_shape There is also a different parametrization in the German Wikipedia article: https://de.wikipedia.org/wiki/Kleinsche_Flasche#Beschreibung_im_dreidimensionalen_Raum .. code-block:: csharp double sinV = Math.Sin(v); double cosV = Math.Cos(v); double sinU = Math.Sin(u); double cosU = Math.Cos(u); double cosU2 = cosU * cosU; double cosU3 = cosU2 * cosU; double cosU4 = cosU3 * cosU; double cosU5 = cosU4 * cosU; double cosU6 = cosU5 * cosU; double cosU7 = cosU6 * cosU; x = -2.0 / 15 * cosU * (3 * cosV - 30 * sinU + 90 * cosU4 * sinU - 60 * cosU6 * sinU + 5 * cosU * cosV * sinU); y = -1.0 / 15 * sinU * (3 * cosV - 3 * cosU2 * cosV - 48 * cosU4 * cosV + 48 * cosU6 * cosV - 60 * sinU + 5 * cosU * cosV * sinU - 5 * cosU3 * cosV * sinU - 80 * cosU5 * cosV * sinU + 80 * cosU7 * cosV * sinU); z = 2.0 / 15 * (3 + 5 * cosU * sinU) * sinV; // Note: Move y up a bit and invert. // Invert x to orient the "outer" parts of the bottle outwardly. // If you don't use a BackMaterial, then parts inside the opening are culled. double a = 1.5; x = a * (-x); y = a * (2 - y); z = a * (z); |TestKleinBottle2_a| `\quad` |TestKleinBottle2_b| .. |TestKleinBottle2_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBottle2_a.3D.xml.jpg :width: 30 % .. |TestKleinBottle2_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBottle2_a.3D.xml.jpg :width: 30 % **Left figure**: Klein bottle, version 2 (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Klein bottle, version 2 (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Klein bottle, version 1 ------------------------------------ See also: http://www.mapleprimes.com/maplesoftblog/95570-Klein-Bottle-Plot See also: http://www.chebfun.org/examples/geom/ParametricSurfaces.html See also: https://mathworld.wolfram.com/KleinBottle.html .. code-block:: csharp x = (3 * (1 + Math.Sin(v)) + 2 * (1 - Math.Cos(v) / 2) * Math.Cos(u)) * Math.Cos(v); y = (-2 * (1 - Math.Cos(v) / 2) * Math.Sin(u)); z = (4 + 2 * (1 - Math.Cos(v) / 2) * Math.Cos(u)) * Math.Sin(v); |TestKleinBottle_a| `\quad` |TestKleinBottle_b| .. |TestKleinBottle_a| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBottle_a.3D.xml.jpg :width: 30 % .. |TestKleinBottle_b| image:: ../_static/ParametricSurfaces/Nonorientable/TestKleinBottle_a.3D.xml.jpg :width: 30 % **Left figure**: Klein bottle, version 1 (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Klein bottle, version 1 (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| 3D Boy Surface, version 1 ----------------------------------- See also: https://en.wikipedia.org/wiki/Boy%27s_surface See also: http://mathworld.wolfram.com/BoySurface.html .. code-block:: csharp double sqrt5 = Math.Sqrt(5); // w = u*e^(iv) double wr = Math.Cos(v); double wi = Math.Sin(v); Complex w = u * new Complex(wr, wi); Complex w3 = w * w * w; Complex w4 = w3 * w; Complex w6 = w3 * w3; Complex d = w6 + sqrt5 * w3 - 1; Complex wa = w * (1 - w4) / d; Complex wb = w * (1 + w4) / d; Complex wc = (1 + w6) / d; double g1 = -1.5 * wa.Imaginary; double g2 = -1.5 * wb.Real; double g3 = wc.Imaginary - 0.5; double l2 = g1 * g1 + g2 * g2 + g3 * g3; x = g1 / l2; y = -g2 / l2; z = g3 / l2; |picTestBoy1a| `\quad` |picTestBoy1b| .. |picTestBoy1a| image:: ../_static/ParametricSurfaces/Nonorientable/TestBoy.3D.xm.jpg :width: 30 % .. |picTestBoy1b| image:: ../_static/ParametricSurfaces/Nonorientable/TestBoy.3D.xml.jpg :width: 30 % **Left figure**: parametric plot of the Boy Surface (parametrization 1). Camera angles are `\theta=135^\circ`. **Reft figure**: parametric plot of the Boy Surface (parametrization 1). Camera angles are `\theta=135^\circ`. |newpage| 3D Boy Surface, version 2 ------------------------------------------- See also: https://en.wikipedia.org/wiki/Boy%27s_surface See also: http://mathworld.wolfram.com/BoySurface.html .. code-block:: csharp double sqrt2 = Math.Sqrt(2); double s2v = Math.Sin(2 * v); double cu = Math.Cos(u); double cv = Math.Cos(v); double cv2 = cv * cv; double n1 = sqrt2 * cv2 * Math.Cos(2*u); double xn = sqrt2 * cv2 * Math.Cos(2 * u) + cu * s2v; double yn = sqrt2 * cv2 * Math.Sin(2 * u) - Math.Sin(u) * s2v; double zn = 3 * cv2; double d = 2 - sqrt2 * Math.Sin(3 * u) * s2v; x = xn / d; y = yn / d; z = zn / d; |picTestBoy2a| `\quad` |picTestBoy2b| .. |picTestBoy2a| image:: ../_static/ParametricSurfaces/Nonorientable/TestBoySurface2_a.3D.xml.jpg :width: 30 % .. |picTestBoy2b| image:: ../_static/ParametricSurfaces/Nonorientable/TestBoySurface2_b.3D.xml.jpg :width: 30 % **Left figure**: parametric plot of the Boy Surface (parametrization 1). Camera angles are `\theta=135^\circ`. **Reft figure**: parametric plot of the Boy Surface (parametrization 1). Camera angles are `\theta=135^\circ`. |newpage| Morin Surface ------------------------------------------- // See also: http://www.3d-meier.de/tut3/Seite221.html // Morin Surface // See also: https://mathcurve.com/surfaces.gb/morin/morin.shtml // See also: https://en.wikipedia.org/wiki/Morin_surface // See also: Bednorz 2019 .. code-block:: csharp var k = 1.0; var n = 3.0; var Sqrt2 = Math.Sqrt(2); var cu = Math.Cos(u); var su = Math.Sin(u); var K = cu / (Sqrt2 - k * Math.Sin(2 * u) * Math.Sin(n * v)); var x = K * (2 / (n - 1) * cu * Math.Cos((n - 1) * v) + Sqrt2 * su * Math.Cos(v)); var y = K * (2 / (n - 1) * cu * Math.Sin((n - 1) * v) - Sqrt2 * su * Math.Sin(v)); var z = K * cu; |11a_TestMorin3| `\quad` |12a_TestMorin5| `\quad` |13a_TestMorin9| .. |11a_TestMorin3| image:: ../_static/ParametricSurfaces/Nonorientable/11a_TestMorin3.3D.xml.jpg :width: 30 % .. |12a_TestMorin5| image:: ../_static/ParametricSurfaces/Nonorientable/12a_TestMorin5.3D.xml.jpg :width: 30 % .. |13a_TestMorin9| image:: ../_static/ParametricSurfaces/Nonorientable/13a_TestMorin9.3D.xml.jpg :width: 30 % **Left figure**: Morin Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Middle figure**: Morin Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Morin Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |11b_TestMorin3| `\quad` |12b_TestMorin5| `\quad` |13b_TestMorin9| .. |11b_TestMorin3| image:: ../_static/ParametricSurfaces/Nonorientable/11b_TestMorin3.3D.xml.jpg :width: 30 % .. |12b_TestMorin5| image:: ../_static/ParametricSurfaces/Nonorientable/12b_TestMorin5.3D.xml.jpg :width: 30 % .. |13b_TestMorin9| image:: ../_static/ParametricSurfaces/Nonorientable/13b_TestMorin9.3D.xml.jpg :width: 30 % **Left figure**: Morin Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Middle figure**: Morin Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Morin Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`).