Nonorientable (one-sided) Surfaces

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

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

Left figure: Moebius Strip (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Moebius Strip (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

Cross-Cap Surface#

See also: https://mathworld.wolfram.com/Cross-Cap.html

See also Wikipedia [1484], MathWorld [1160].

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

Left figure: Cross-Cap Surface (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Cross-Cap Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

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 [1484], MathWorld [1160].

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

Left figure: Pseudo Cross-Cap Surface (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Pseudo Cross-Cap Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

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/

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 [1484], MathWorld [1160].

TestRoman_a \(\quad\) TestRoman_b

Left figure: Roman Surface (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Roman Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

Klein bagel#

See also: https://en.wikipedia.org/wiki/Klein_bottle#The_figure_8_immersion

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

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

Left figure: Klein bagel (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Klein bagel (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

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

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

Left figure: Klein bottle, version 3 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Klein bottle, version 3 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

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

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

Left figure: Klein bottle, version 2 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Klein bottle, version 2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

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

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

Left figure: Klein bottle, version 1 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Klein bottle, version 1 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

3D Boy Surface, version 1#

See also: https://en.wikipedia.org/wiki/Boy%27s_surface

See also: http://mathworld.wolfram.com/BoySurface.html

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

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 \(\theta=135^\circ\).

3D Boy Surface, version 2#

See also: https://en.wikipedia.org/wiki/Boy%27s_surface

See also: http://mathworld.wolfram.com/BoySurface.html

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

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 \(\theta=135^\circ\).

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

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

Left figure: Morin Surface (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Middle figure: Morin Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Morin Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

11b_TestMorin3 \(\quad\) 12b_TestMorin5 \(\quad\) 13b_TestMorin9

Left figure: Morin Surface (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]).

Middle figure: Morin Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

Right figure: Morin Surface (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]).

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