.. |newpage| raw:: latex \newpage .. |br| raw:: html
|newpage| Minimal surfaces ============================================== Helicoid ------------------------------------- See also: https://mathworld.wolfram.com/Helicoid.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |TestHelicoid_a| `\quad` |TestHelicoid_b| .. |TestHelicoid_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestHelicoid_a.3D.xml.jpg :width: 30 % .. |TestHelicoid_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestHelicoid_a.3D.xml.jpg :width: 30 % **Left figure**: Helicoid (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Helicoid (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). .. code-block:: csharp double a = 1; x = u * Math.Cos(v); z = u * Math.Sin(v); y = v; References Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 449 and 644, 1997. |newpage| Bours minimal surface ------------------------------------------------ See also: https://mathworld.wolfram.com/BoursMinimalSurface.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); double c32u = Math.Cos(1.5*v); double cv = Math.Cos(v); double c2v = Math.Cos(2*v); double u2 = 0.5*u*u; double u32 = (4/3)* Math.Sqrt(u*u*u); x = u * cv - u2 * c2v; y = -u * sv - u2 * s2v; z = u32 * c32u; |TestBour_a| `\quad` |TestBour_b| .. |TestBour_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestBour_a.3D.xml.jpg :width: 30 % .. |TestBour_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestBour_a.3D.xml.jpg :width: 30 % **Left figure**: Bours Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Bours Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Catalan minimal surface -------------------------------------------------- See also: https://mathworld.wolfram.com/CatalanMinimalSurface.html See also: https://en.wikipedia.org/wiki/Catalan%27s_minimal_surface See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. .. code-block:: csharp x = u - u * u * u / 3 + u * v * v; y = u * u - v * v; z = v - v * v * v / 3 + v * u * u; |TestCatalan_a| `\quad` |TestCatalan_b| .. |TestCatalan_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestCatalan_a.3D.xml.jpg :width: 30 % .. |TestCatalan_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestCatalan_a.3D.xml.jpg :width: 30 % **Left figure**: Catalan Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Catalan Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Ennepers first minimal surface ------------------------------------------------------- See also: https://mathworld.wolfram.com/EnnepersMinimalSurface.html See also: https://en.wikipedia.org/wiki/Enneper_surface See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. .. code-block:: csharp x = u - u * u * u / 3 + u * v * v; y = u * u - v * v; z = v - v * v * v / 3 + v * u * u; |TestEnneper_a| `\quad` |TestEnneper_b| .. |TestEnneper_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestEnneper_a.3D.xml.jpg :width: 30 % .. |TestEnneper_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestEnneper_a.3D.xml.jpg :width: 30 % **Left figure**: Ennepers Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Ennepers Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Enneper second minimal surface ------------------------------------------------------ See also: https://mathcurve.com/surfaces.gb/enneper/enneper.shtml See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. .. code-block:: csharp double n = 2; double a = 1; Complex i = Complex.ImaginaryOne; Complex w = new Complex(u, v); Complex w2nm1 = Complex.Pow(w, 2*n-1)/(2*n-1); Complex wn = Complex.Pow(w, n); x = a * (w - w2nm1).Real; y = a * (-i*(w + w2nm1)).Real; z = 2 * a * (wn/n).Real; |TestEnneper2_a| `\quad` |TestEnneper2_b| .. |TestEnneper2_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestEnneper2_a.3D.xml.jpg :width: 30 % .. |TestEnneper2_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestEnneper2_a.3D.xml.jpg :width: 30 % **Left figure**: Ennepers Minimal Surface2 (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Ennepers Minimal Surface2 (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Hennebergs minimal surface ------------------------------------------------ See also: https://mathworld.wolfram.com/HennebergsMinimalSurface.html See also: https://en.wikipedia.org/wiki/Henneberg_surface See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. .. code-block:: csharp double sv = Math.Sin(v); double s3v = Math.Sin(3*v); double cv = Math.Cos(v); double c2v = Math.Cos(2*v); double c3v = Math.Cos(3*v); double shu = Math.Sinh(u); double ch2u = Math.Cosh(2*u); double sh3u = Math.Sinh(3*u); x = 2*shu * cv - (2.0/3.0)*sh3u * c3v; y = 2*shu * sv + (2.0/3.0)*sh3u * s3v; z = 2*ch2u * c2v; |TestHenneberg_a| `\quad` |TestHenneberg_b| .. |TestHenneberg_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestHenneberg_a.3D.xml.jpg :width: 30 % .. |TestHenneberg_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestHenneberg_a.3D.xml.jpg :width: 30 % **Left figure**: Hennebergs Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Hennebergs Minimal Surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Scherk’s first minimal surface ------------------------------------------------- See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. See also https://mathcurve.com/surfaces.gb/scherk/scherk.shtml See also :cite:t:`Krivoshapko2015`, p. 431 .. code-block:: csharp var a = 1 / Math.PI; var x = u; var z = v; var y = a * Math.Log(Math.Cos(v / a) / Math.Cos(u / a)); z = -z; |TestScherk1_a| `\quad` |TestScherk1_b| .. |TestScherk1_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestScherk1_a.3D.xml.jpg :width: 30 % .. |TestScherk1_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestScherk1_a.3D.xml.jpg :width: 30 % **Left figure**: Scherk’s first minimal surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Scherk’s first minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Scherk’s second minimal surface -------------------------------------------------- See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. See also https://mathworld.wolfram.com/ScherksMinimalSurfaces.html See also :cite:t:`Krivoshapko2015`, p. 442 .. code-block:: csharp var t = v; var r = u; var r2 = r * r; var ct = Math.Cos(t); var st = Math.Sin(t); var x = Math.Log((1 + r2 + 2 * r * ct) / (1 + r2 - 2 * r * ct)); var y = Math.Log((1 + r2 - 2 * r * st) / (1 + r2 + 2 * r * st)); var z = 2 * Math.Atan((2 * r2 * Math.Sin(2 * t)) / (r2 * r2 - 1)); y = -y; |TestScherk2_a| `\quad` |TestScherk2_b| .. |TestScherk2_a| image:: ../_static/ParametricSurfaces/RuledMinimal/TestScherk2_a.3D.xml.jpg :width: 30 % .. |TestScherk2_b| image:: ../_static/ParametricSurfaces/RuledMinimal/TestScherk2_a.3D.xml.jpg :width: 30 % **Left figure**: Scherk’s second minimal surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Scherk’s second minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Richmond minimal surface ------------------------------------------------------------------------- // See also: http://www.3d-meier.de/tut3/Seite250.html // Richmond Surface III // See also: https://en.wikipedia.org/wiki/Richmond_surface .. code-block:: csharp var n = 2.0; var u2s = Math.Pow(u, 2 * n + 1) / (4 * n + 2); var x = -Math.Cos(v) / (2 * u) - u2s * Math.Cos(-(2 * n + 1) * v); var y = -Math.Sin(v) / (2 * u) + u2s * Math.Sin(-(2 * n + 1) * v); var z = Math.Pow(u, n) * Math.Cos(n * v) / n; |TestRichmond_a| `\quad` |TestRichmond_b| `\quad` |TestRichmond_c| .. |TestRichmond_a| image:: ../_static/ParametricSurfaces/RuledMinimal/11a_TestRichmond.3D.xml.jpg :width: 30 % .. |TestRichmond_b| image:: ../_static/ParametricSurfaces/RuledMinimal/11b_TestRichmond.3D.xml.jpg :width: 30 % .. |TestRichmond_c| image:: ../_static/ParametricSurfaces/RuledMinimal/11c_TestRichmond.3D.xml.jpg :width: 30 % **Left figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Middle figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |TestRichmond_d| `\quad` |TestRichmond_e| `\quad` |TestRichmond_f| .. |TestRichmond_d| image:: ../_static/ParametricSurfaces/RuledMinimal/11d_TestRichmond.3D.xml.jpg :width: 30 % .. |TestRichmond_e| image:: ../_static/ParametricSurfaces/RuledMinimal/11e_TestRichmond.3D.xml.jpg :width: 30 % .. |TestRichmond_f| image:: ../_static/ParametricSurfaces/RuledMinimal/11f_TestRichmond.3D.xml.jpg :width: 30 % **Left figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Middle figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |newpage| Generalized Ennneper surfaces ------------------------------------------------------------------------- // See also: http://www.3d-meier.de/tut3/Seite247.html // Wavy Enneper Surface .. code-block:: csharp var s = 2.0; var u2s = Math.Pow(u, 2 * s - 1) / (2 * s - 1); var x = u * Math.Cos(v) - u2s * Math.Cos((2 * s - 1) * v); var y = -u * Math.Sin(v) - u2s * Math.Sin((2 * s - 1) * v); var z = 2 * Math.Pow(u, s) * Math.Cos(s * v) / s; |TestGenEnneper_b| `\quad` |TestGenEnneper_c| `\quad` |TestGenEnneper_d| .. |TestGenEnneper_b| image:: ../_static/ParametricSurfaces/RuledMinimal/12b_TestGenEnneper.3D.xml.jpg :width: 30 % .. |TestGenEnneper_c| image:: ../_static/ParametricSurfaces/RuledMinimal/12c_TestGenEnneper.3D.xml.jpg :width: 30 % .. |TestGenEnneper_d| image:: ../_static/ParametricSurfaces/RuledMinimal/12d_TestGenEnneper.3D.xml.jpg :width: 30 % **Left figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Middle figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Richmond minimal surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). |TestGenEnneper_e| `\quad` |TestGenEnneper_f| `\quad` |TestGenEnneper_g| .. |TestGenEnneper_e| image:: ../_static/ParametricSurfaces/RuledMinimal/12e_TestGenEnneper.3D.xml.jpg :width: 30 % .. |TestGenEnneper_f| image:: ../_static/ParametricSurfaces/RuledMinimal/12f_TestGenEnneper.3D.xml.jpg :width: 30 % .. |TestGenEnneper_g| image:: ../_static/ParametricSurfaces/RuledMinimal/12g_TestGenEnneper.3D.xml.jpg :width: 30 % **Left figure**: Generalized Ennneper surface (see also Wikipedia :cite:p:`Wikipedia2D010`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Middle figure**: Generalized Ennneper surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`). **Right figure**: Generalized Ennneper surface (see also Wikipedia :cite:p:`Wikipedia2D011`, :cite:t:`Gray2006`, :cite:t:`Krivoshapko2015`).