Generalisations of common surfaces

Generalisations of common surfaces#

Parametric Surfaces: \(x=f(u,v),\) \(y=g(u,v),\) \(z=h(u,v)\)

Ellipsoid#

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

The parametric equations of an ellipsoid can be written as

\[x(u, v) = a \cos(u) \sin(v),\]
\[y(u, v) = b \sin(u) \sin(v),\]
\[z(u, v) = r \sin(t),\]

for \(u \in [0, 2 \pi)\) and \(v \in [0, \pi]\).

See also: http://paulbourke.net/geometry/spherical/ See also: https://mathcurve.com/surfaces.gb/lame/lame.shtml See also: https://mathworld.wolfram.com/Ellipsoid.html

See also Gray et al. [353], Krivoshapko and Ivanov [399].

var a = 1.0;
var b = 2.0;
var c = 3.0;
var x = a * Math.Cos(u) * Math.Cos(v);
var y = b * Math.Cos(u) * Math.Sin(v);
var z = c * Math.Sin(u);

If the lengths of two axes of an ellipsoid are the same, the figure is called an ellipsoid of revolution or spheroid. Denote the equal semi-axes lengths of a spheroid \(a=b\), call \(a\) the equatorial radius, and call the other semi-axis length the polar radius \(c\). Then if \(a>c\), the spheroid is called an oblate spheroid, and if \(a<c\), the spheroid is called an prolate spheroid. If all three semi-axes lengths are the same so \(a=b=c\), the ellipsoid is a sphere.

01a_Ellipsoid.3D \(\quad\) 01b_Ellipsoid.3D \(\quad\) 01c_Ellipsoid.3D

Superellipsoid#

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

Superellipsoid is the name given to a family of shapes formed from the spherical product of two superquadratric curves. These shapes can be used to model a wide range of shapes including spheres, cylinders, and parallelepipeds as well as shapes in between. The parametric equations of an superellipsoid can be written as

\[x(u, v) = \mathrm{sgn}(\cos(u)) \cdot |\cos(u)| ^{p1} \cdot \mathrm{sgn}(\cos(v)) \cdot |\cos(v)|^{p2},\]
\[y(u, v) = \mathrm{sgn}(\cos(u)) \cdot |\cos(u)| ^{p1} \cdot \mathrm{sgn}(\sin(v)) \cdot |\sin(v)|^{p2},\]
\[z(u, v) = \mathrm{sgn}(\sin(u)) \cdot |\sin(u)|^{p1},\]

where \(\displaystyle \frac{-\pi}{2} \le u \le \frac{\pi}{2}\), \(-\pi \le v \le \pi\), and \(0 < p1, p2 < \infty\).

double p1 = 2.0;
double p2 = 3.8;

double u = u;
double v = v;

\cos(u) = Math.Cos(u);
\cos(v) = Math.Cos(v);
\sin(u) = Math.Sin(u);
\sin(v) = Math.Sin(v);

tmp  = Math.Sign(\cos(u)) * Math.Pow(Math.Abs(\cos(u)),p1);
x = tmp * Math.Sign(\cos(v)) * Math.Pow(Math.Abs(\cos(v)),p2);
y = -Math.Sign(\sin(u)) * Math.Pow(Math.Abs(\sin(u)),p1);
z = tmp * Math.Sign(\sin(v)) * Math.Pow(Math.Abs(\sin(v)),p2);

02a_SuperEllipse.3D \(\quad\) 02b_SuperEllipse.3D \(\quad\) 02c_SuperEllipse.3D

Hexaedron#

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

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

double cosu = Math.Cos(u);
double sinu = Math.Sin(u);
double cosv = Math.Cos(v);
double sinv = Math.Sin(v);
x = cosv * cosv * cosv * cosu * cosu * cosu;
y = -sinu * sinu * sinu;
z = sinv * sinv * sinv * cosu * cosu * cosu;

03a_Hexaedron.3D \(\quad\) 03b_Hexaedron.3D \(\quad\) 03c_Hexaedron.3D

Super Toroid#

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

See also: http://paulbourke.net/geometry/toroidal/

04a_SuperToroid.3D \(\quad\) 04b_SuperToroid.3D \(\quad\) 04c_SuperToroid.3D

double r0 = 1;
double r1 = 0.3;

double tmp;
double ct1,ct2,st1,st2;

double p1 = 2.0;
double p2 = 3.8;
double t1 = u;
double t2 = v;

ct1 = Math.Cos(t1);
ct2 = Math.Cos(t2);
st1 = Math.Sin(t1);
st2 = Math.Sin(t2);

tmp  = r0 + r1 * Math.Sign(ct2) * Math.Pow(Math.Abs(ct2),p2);

x = tmp * Math.Sign(ct1) * Math.Pow(Math.Abs(ct1),p1);
y = -tmp * Math.Sign(st1) * Math.Pow(Math.Abs(st1),p1);
z = r1 * Math.Sign(st2) * Math.Pow(Math.Abs(st2),p2);

Elliptic 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 D05a_EllipticHelicoid.3D.xml (D05a-c).

See also: https://mathworld.wolfram.com/EllipticHelicoid.html

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

05a_EllipticHelicoid.3D \(\quad\) 05b_EllipticHelicoid.3D \(\quad\) 05c_EllipticHelicoid.3D

double a = 0.5;
double b = 1.5;
double c = 1;
double su = Math.Sin(u);
double cu = Math.Cos(u);

x = a * v * cu;
z = b * v * su;
y = c * u;

Hyperbolic 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 D06a_Hyperhelicoid.3D.xml (D06a-c).

The example below uses the following code in C#

var x = (Math.Sinh(v) * Math.Cos(3 * u)) / (1 + Math.Cosh(u) * Math.Cosh(v));
var y = (Math.Sinh(v) * Math.Sin(3 * u)) / (1 + Math.Cosh(u) * Math.Cosh(v));
var z = (Math.Cosh(v) * Math.Sinh(u)) / (1 + Math.Cosh(u) * Math.Cosh(v));

See also: https://mathworld.wolfram.com/HyperbolicHelicoid.html

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

06a_Hyperhelicoid.3D \(\quad\) 06b_Hyperhelicoid.3D \(\quad\) 06c_Hyperhelicoid.3D

Lemniscate#

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

See also: http://paulbourke.net/geometry/lemniscape/

var x = Math.Cos(v) * Math.Sqrt(Math.Abs(Math.Sin(2 * u))) * Math.Cos(u);
var y = Math.Cos(v) * Math.Sqrt(Math.Abs(Math.Sin(2 * u))) * Math.Sin(u);
var z = x * x - y * y + 2 * x * y * Math.Tan(v) * Math.Tan(v);

07a_Lemnescate.3D \(\quad\) 07b_Lemnescate.3D \(\quad\) 07c_Lemnescate.3D

Bohemian Dome#

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

See also: https://mathworld.wolfram.com/BohemianDome.html

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

var a = 0.5;
var b = 1.5;
var c = 1;
var su = Math.Sin(u);
var sv = Math.Sin(v);
var cu = Math.Cos(u);
var cv = Math.Cos(v);

var x = a * cu;
var z = b * cv + a * su;
var y = c * sv;

08a_BohemianDome.3D \(\quad\) 08b_BohemianDome.3D \(\quad\) 08c_BohemianDome.3D

Dupin1 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_Dupin1.3D.xml (D09a-c).

The example below uses the following code in C#

var a = 1.5;
var b = 1.4;
var c = Math.Sqrt(a * a - b * b);
var d = b / 2;
var su = Math.Sin(u);
var sv = Math.Sin(v);
var cu = Math.Cos(u);
var cv = Math.Cos(v);
var den = a - c * cu * cv;

var x = (b * sv * (c * cu - d)) / den;
var y = (d * (c - a * cu * cv) + b * b * cu) / den;
var z = (b * su * (a - d * cv)) / den;

See also: https://mathcurve.com/surfaces.gb/cycliddedupin/cyclidededupin.shtml

See also: https://en.wikipedia.org/wiki/Dupin_cyclide#Elliptic_cyclides

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

09a_Dupin1.3D \(\quad\) 09b_Dupin1.3D \(\quad\) 09c_Dupin1.3D

Dupin2 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_Dupin2.3D.xml (D10a-c).

The example below uses the following code in C#

var p = 2;
var k = 0.7;
var den = 1 + u * u + v * v;
var x = 0.5 * p * (2 * v * v + k * (1 - u * u - v * v)) / den;
var y = p * u * (v * v + k) / den;
var z = p * v * (1 + u * u - k) / den;

See also: https://mathcurve.com/surfaces.gb/cycliddedupin/cyclidededupin.shtml

See also: https://en.wikipedia.org/wiki/Dupin_cyclide#Parabolic_cyclides

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

10a_Dupin2.3D \(\quad\) 10b_Dupin2.3D \(\quad\) 10c_Dupin2.3D

Dinis Surface (twisted pseudosphere)#

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_Dini.3D.xml (D11a-c).

The example below uses the following code in C#

var a = 1;
var b = 0.2;
var x = a * Math.Cos(u) * Math.Sin(v);
var y = a * Math.Sin(u) * Math.Sin(v);
var z = a * (Math.Cos(v) + Math.Log(Math.Tan(0.5 * v))) + b * u;

See also: https://mathworld.wolfram.com/DinisSurface.html

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

11a_Dini.3D \(\quad\) 11b_Dini.3D \(\quad\) 11c_Dini.3D

Plueckers conoid#

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 D12a_Pluecker.3D.xml (D12a-f).

// See : https://mathworld.wolfram.com/PlueckersConoid.html // See : Gray, p. 436 // See : https://en.wikipedia.org/wiki/Pl%C3%BCcker%27s_conoid

var n = 2;
var r = u;
var theta = v;

var x = r * Math.Cos(theta);
var y = r * Math.Sin(theta);
var z = Math.Sin(n * theta);

12a_Pluecker.3D \(\quad\) 12b_Pluecker.3D \(\quad\) 12c_Pluecker.3D

12d_Pluecker.3D \(\quad\) 12e_Pluecker.3D \(\quad\) 12f_Pluecker.3D

References

Gray, A. “Plücker’s Conoid.” Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 435-437, 1997.

Umbilic Torus#

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 D13a_UmbilicTorus.3D.xml (D13a-c).

var x = Math.Cos(u) * (7 + Math.Cos(u / 3 - 2 * v) + 2 * Math.Cos(u / 3 + v));
var y = Math.Sin(u) * (7 + Math.Cos(u / 3 - 2 * v) + 2 * Math.Cos(u / 3 + v));
var z = Math.Sin(u / 3 - 2 * v) + 2 * Math.Sin(u / 3 + v);

// See also: http://www.3d-meier.de/tut3/Seite61.html // Umbilic Torus

13a_UmbilicTorus.3D \(\quad\) 13b_UmbilicTorus.3D \(\quad\) 13c_UmbilicTorus.3D

Skidian’s ruled 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 D14a_Skidan.3D.xml (D14a-f).

var a = Math.PI / 4.0;
var h = 2.0;
var n = 4.0;

var b = h * Math.Abs(Math.Cos(n * v));
var ca = Math.Cos(a);
var sa = Math.Sin(a);

var x = (u * sa + b * ca) * Math.Sin(v);
var y = (u * sa + b * ca) * Math.Cos(v);
var z = (u * ca - b * sa);

// See also: http://www.3d-meier.de/tut3/Seite227.html // Skidan Ruled Surface // See Krivoshapko, p. 499

14a_Skidan.3D \(\quad\) 14b_Skidan.3D \(\quad\) 14c_Skidan.3D

14d_Skidan.3D \(\quad\) 14e_Skidan.3D \(\quad\) 14f_Skidan.3D

Umbrella 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 D15a_Umbrella.3D.xml (D15a-f).

var R = 0.6;
var h = 0.6;
var n = 8.0;

var r = R / n;
var x = ctx.cbrt(u) * ((R - r)) * Math.Cos(v) + r * Math.Cos((n - 1) * v);
var y = ctx.cbrt(u) * ((R - r)) * Math.Sin(v) - r * Math.Sin((n - 1) * v);
var z = h * (1 - u);

// See also: http://www.3d-meier.de/tut3/Seite215.html // Umbrella Surface // See Krivoshapko, p. 507 - 509 // See Krivoshapko, p. 513 - 515 // See Krivoshapko, p. 521, 526, 530, 531, 533

15a_Umbrella.3D \(\quad\) 15b_Umbrella.3D \(\quad\) 15c_Umbrella.3D

15d_Umbrella.3D \(\quad\) 15e_Umbrella.3D \(\quad\) 15f_Umbrella.3D

Cyclic surfaces (generalized torus)#

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 D16a_CyclicSurface1.3D.xml (D16a-n).

var a = 1.0;
var b = 2.0;
var d = 0.0;
var p = 1.0;

var R = a * (1 - d * Math.Cos(p * u));
var x = (b + R * Math.Cos(v)) * Math.Cos(u);
var y = (b + R * Math.Cos(v)) * Math.Sin(u);
var z = R * Math.Sin(v);

// See Krivoshapko, p. 376

16a_CyclicSurface1.3D \(\quad\) 16b_CyclicSurface2.3D \(\quad\) 16c_CyclicSurface1.3D

16d_CyclicSurface2.3D \(\quad\) 16e_CyclicSurface2.3D \(\quad\) 16f_CyclicSurface2.3D

16g_CyclicSurface2.3D \(\quad\) 16h_CyclicSurface2.3D \(\quad\) 16i_CyclicSurface2.3D

16j_CyclicSurface2.3D \(\quad\) 16k_CyclicSurface2.3D \(\quad\) 16l_CyclicSurface3.3D

16m_CyclicSurface3.3D \(\quad\) 16n_CyclicSurface3.3D

Goursat 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 D17a_Goursat1.3D.xml (D17a-f).

The example below uses the following code in C#

var a = -0.33;
var b = 0.2;
var c = -3;
var p = 1;

var su = Math.Sin(u);
var cu = Math.Cos(u);
var sv = Math.Sin(v);
var cv = Math.Cos(v);

var su4 = su * su * su * su;
var cu4 = cu * cu * cu * cu;

var sv4 = sv * sv * sv * sv;
var cv4 = cv * cv * cv * cv;

var D = (su4 + cu4) * cv4 + sv4;
var R = Math.Sqrt((-b + p * Math.Sqrt(b * b - 4 * c * (a + D))) / (2 * (a + D)));
var x = R * cv * cu;
var y = R * cv * su;
var z = R * sv;

// See : Krivoshapko (2015), p. 643 // See: https://mathworld.wolfram.com/GoursatsSurface.html // See Gray 1997, p. 314) // See: https://mathcurve.com/surfaces.gb/goursat/goursat.shtml

17a_Goursat1.3D \(\quad\) 17b_Goursat4.3D \(\quad\) 17c_Goursat2.3D

17d_Goursat1.3D \(\quad\) 17e_Goursat4.3D \(\quad\) 17f_Goursat2.3D

Cyclides triples#

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 D18a_CyclidesTriple.3D.xml (D18a-f).

The example below uses the following code in C#

var su = Math.Sin(u);
var s2u = Math.Sin(2 * u);
var cu = Math.Cos(u);
var sv = Math.Sin(v);
var cv = Math.Cos(v);

var x = cv * sv * sv * sv * s2u * cu;
var y = cv * sv * sv * sv * s2u * su;
var z = cv * sv * sv * s2u;

// See : Krivoshapko (2015), p. 651

18a_CyclidesTriple.3D \(\quad\) 18b_CyclidesTriple.3D \(\quad\) 18c_CyclidesTriple.3D

18d_CyclidesTriple.3D \(\quad\) 18e_CyclidesTriple.3D \(\quad\) 18f_CyclidesTriple.3D

Ship Lamé#

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 D19a_ShipLame.3D.xml (D19a-c).

The example below uses the following code in C#

var B = 6.5;
var L = 4.0;
var T = 2.0;

var x = u * L;
var y = Math.Sign(Math.Cos(v)) * B * Math.Sqrt(1 - u * u * u * u) * Math.Sqrt(Math.Abs(Math.Cos(v))) / 2.0;
var z = Math.Sign(Math.Sin(v)) * T * Math.Sqrt(1 - u * u) * Math.Sqrt(Math.Abs(Math.Sin(v)));

// See : Krivoshapko (2015), p. 671

19a_ShipLame.3D \(\quad\) 19b_ShipLame.3D \(\quad\) 19c_ShipLame.3D

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