Torus knots

Torus knots#

See also: https://katlas.org/wiki/36_Torus_Knots

Torus knot 3:2#

Some text

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

See also: https://katlas.org/wiki/T(3,2)

An example in C#

double R = 100;
double r = 25;
double p = 3;
double q = 2;
var x = ((R + r * Math.Cos(p * t)) * Math.Cos(q * t)) / 50;
var y = ((R + r * Math.Cos(p * t)) * Math.Sin(q * t)) / 50;
var z = (r * Math.Sin(p * t)) / 50;

Path_Torus_Knot_3_2_aa \(\quad\) Path_Torus_Knot_3_2_bb \(\quad\) Path_Torus_Knot_3_2_cc

Left figure: Torus knot 3:2 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Torus knot 3:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Torus knot 3:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Torus knot 5:2#

Some text

See also: https://en.wikipedia.org/wiki/Cinquefoil_knot

See also: https://katlas.org/wiki/T(5,2)

An example in C#

double R = 100;
double r = 25;
double p = 5;
double q = 2;
var x = ((R + r * Math.Cos(p * t)) * Math.Cos(q * t)) / 50;
var y = ((R + r * Math.Cos(p * t)) * Math.Sin(q * t)) / 50;
var z = (r * Math.Sin(p * t)) / 50;

Path_Torus_Knot_5_2_aa \(\quad\) Path_Torus_Knot_5_2_bb \(\quad\) Path_Torus_Knot_5_2_cc

Left figure: Torus knot 5:2 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Torus knot 5:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Torus knot 5:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Torus knot 7:2#

Some text

See also: https://en.wikipedia.org/wiki/71_knot

See also: https://katlas.org/wiki/T(7,2)

See also: https://katlas.org/wiki/7_1

An example in C#

double R = 100;
double r = 25;
double p = 7;
double q = 2;
var x = ((R + r * Math.Cos(p * t)) * Math.Cos(q * t)) / 50;
var y = ((R + r * Math.Cos(p * t)) * Math.Sin(q * t)) / 50;
var z = (r * Math.Sin(p * t)) / 50;

Path_Torus_Knot_7_2_aa \(\quad\) Path_Torus_Knot_7_2_bb \(\quad\) Path_Torus_Knot_7_2_cc

Left figure: Torus knot 7:2 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Torus knot 7:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Torus knot 7:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Torus knot 7:3#

Some text

See also: https://katlas.org/wiki/T(7,3)

An example in C#

double R = 100;
double r = 25;
double p = 7;
double q = 3;
var x = ((R + r * Math.Cos(p * t)) * Math.Cos(q * t)) / 50;
var y = ((R + r * Math.Cos(p * t)) * Math.Sin(q * t)) / 50;
var z = (r * Math.Sin(p * t)) / 50;

Path_Torus_Knot_7_3_aa \(\quad\) Path_Torus_Knot_7_3_bb \(\quad\) Path_Torus_Knot_7_3_cc

Left figure: Torus knot 7:3 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Torus knot 7:3 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Torus knot 7:3 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Torus knot 15:2#

Some text

See also: https://katlas.org/wiki/T(15,2)

An example in C#

double R = 100;
double r = 25;
double p = 15;
double q = 2;
var x = ((R + r * Math.Cos(p * t)) * Math.Cos(q * t)) / 50;
var y = ((R + r * Math.Cos(p * t)) * Math.Sin(q * t)) / 50;
var z = (r * Math.Sin(p * t)) / 50;

Path_Torus_Knot_15_2_aa \(\quad\) Path_Torus_Knot_15_2_bb \(\quad\) Path_Torus_Knot_15_2_cc

Left figure: Torus knot 15:2 (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Torus knot 15:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Torus knot 15:2 (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

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