Minimal surfaces#

Catenoid#

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_Catenoid.3D.xml (D01a-c).

The example below uses the following code in C#

var c = 2;
var x = c * Math.Cosh(v / c) * Math.Cos(u);
var y = c * Math.Cosh(v / c) * Math.Sin(u);
var z = v;

\(\quad\) \(\quad\)

Helicoid#

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_Helicoid.3D.xml (D02a-c).

The example below uses the following code in C#

var x = u * Math.Cos(v);
var y = u * Math.Sin(v);
var z = v;

See also Wikipedia [1368], MathWorld [1070].

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 449 and 644, 1997.

Bours minimal surface#

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_Bour.3D.xml (D03a-c).

The example below uses the following code in C#

var sv = Math.Sin(v);
var s2v = Math.Sin(2 * v);
var c32u = Math.Cos(1.5 * v);
var cv = Math.Cos(v);
var c2v = Math.Cos(2 * v);
var u2 = 0.5 * u * u;
var u32 = (4.0 / 3.0) * Math.Sqrt(u * u * u);

var x = u * cv - u2 * c2v;
var y = -u * sv - u2 * s2v;
var z = u32 * c32u;

See also Wikipedia [1368], MathWorld [1070].

\(\quad\) \(\quad\)

Catalan minimal surface#

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_Catalan.3D.xml (D04a-c).

The example below uses the following code in C#

var x = u - Math.Sin(u) * Math.Cosh(v);
var y = 4 * Math.Sin(u / 2) * Math.Sinh(v / 2);
var z = 1 - Math.Cos(u) * Math.Cosh(v);

See also Wikipedia [1368], MathWorld [1070].

\(\quad\) \(\quad\)

Ennepers first minimal surface#

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_Enneper.3D.xml (D05a-c).

The example below uses the following code in C#

var x = u - u * u * u / 3 + u * v * v;
var y = u * u - v * v;
var z = v - v * v * v / 3 + v * u * u;

See also Wikipedia [1368], MathWorld [1070].

\(\quad\) \(\quad\)

Enneper second minimal surface#

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_Enneper2.3D.xml (D06a-c).

The example below uses the following code in C#

var n = 2;
var 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);

var x = a * (w - w2nm1).Real;
var y = a * (-i * (w + w2nm1)).Real;
var z = 2 * a * (wn / n).Real;

See also Wikipedia [1368], MathWorld [1070].

\(\quad\) \(\quad\)

Hennebergs minimal surface#

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_Henneberg.3D.xml (D07a-c).

The example below uses the following code in C#

var sv = Math.Sin(v);
var s3v = Math.Sin(3 * v);
var cv = Math.Cos(v);
var c2v = Math.Cos(2 * v);
var c3v = Math.Cos(3 * v);
var shu = Math.Sinh(u);
var ch2u = Math.Cosh(2 * u);
var sh3u = Math.Sinh(3 * u);

var x = 2 * shu * cv - (2.0 / 3.0) * sh3u * c3v;
var y = 2 * shu * sv + (2.0 / 3.0) * sh3u * s3v;
var z = 2 * ch2u * c2v;

See also Wikipedia [1368], MathWorld [1070].

\(\quad\) \(\quad\)

Scherk’s first minimal surface#

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 D08a_Scherk1.3D.xml (D08a-c).

The example below uses the following code in C#

var a = 1 / Math.PI;
var y = u;
var x = a * Math.Log(Math.Cos(v / a) / Math.Cos(u / a));
var z = v;

See also Wikipedia [1368], MathWorld [1070].

See also https://mathcurve.com/surfaces.gb/scherk/scherk.shtml See also Krivoshapko and Ivanov [399], p. 431

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Scherk’s second minimal surface#

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 D09a_Scherk2.3D.xml (D09a-c).

The example below uses the following code in C#

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;

See also Wikipedia [1368], MathWorld [1070].

See also Krivoshapko and Ivanov [399], p. 442

\(\quad\) \(\quad\)

Richmond minimal surface#

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 D10a_Richmond.3D.xml (D10a-f).

The example below uses the following code in C#

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;

// See also: http://www.3d-meier.de/tut3/Seite250.html // Richmond Surface III // See also: https://en.wikipedia.org/wiki/Richmond_surface

\(\quad\) \(\quad\)

Generalized Ennneper surfaces#

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 D11a_GenEnneper.3D.xml (D11a-f).

The example below uses the following code in C#

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;