General knots

Trefoil A

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

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

An example in C#

var x = (2 + Math.Cos(3 * t)) * Math.Cos(2 * t);
var y = (2 + Math.Cos(3 * t)) * Math.Sin(2 * t);
var z = Math.Sin(3 * t);

Some text

\(\quad\)

Left figure: Trefoil A. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Trefoil A. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Trefoil B

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

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

An example in C#

var x = (-10 * Math.Cos(t) - 2 * Math.Cos(5 * t) + 15 * Math.Sin(2 * t)) / 10;
var y = (-15 * Math.Cos(2 * t) + 10 * Math.Sin(t) - 2 * Math.Sin(5 * t)) / 10;
var z = (10 * Math.Cos(3 * t)) / 10;

Some text

\(\quad\)

Left figure: Trefoil B. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Trefoil B. Trefoil B. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 180^\circ\).

Square knot

See also: https://en.wikipedia.org/wiki/Square_knot_(mathematics)

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

See also: https://mathcurve.com/courbes3d.gb/plat.vache/plat_vache.shtml

An example in C#

var x = (-22 * Math.Cos(t) - 128 * Math.Sin(t) - 44 * Math.Cos(3 * t) - 78 * Math.Sin(3 * t)) / 10;
var y = (11 * Math.Cos(t) - 43 * Math.Cos(3 * t) + 34 * Math.Cos(5 * t) - 39 * Math.Sin(5 * t)) / 10;
var z = (70 * Math.Cos(3 * t) - 40 * Math.Sin(3 * t) + 8 * Math.Cos(5 * t) - 9 * Math.Sin(5 * t)) / 10;

Some text

\(\quad\)

Left figure: Trefoil A. Perspective camera. Camera angles are \(\theta=0^\circ\) and \(\phi = 180^\circ\).

Right figure: Trefoil B. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 135^\circ\).

Some text

\(\quad\)

Left figure: Trefoil A. Orthographic camera. Camera angles are \(\theta=0^\circ\) and \(\phi = 180^\circ\).

Right figure: Trefoil B. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 135^\circ\).

Granny knot

See also: https://en.wikipedia.org/wiki/Granny_knot_(mathematics)

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

See also: https://mathcurve.com/courbes3d.gb/plat.vache/plat_vache.shtml

An example in C#

var x = (-22 * Math.Cos(t) - 128 * Math.Sin(t) - 44 * Math.Cos(3 * t) - 78 * Math.Sin(3 * t)) / 10;
var y = (-10 * Math.Cos(2 * t) - 27 * Math.Sin(2 * t) + 38 * Math.Cos(4 * t) + 46 * Math.Sin(4 * t)) / 10;
var z = (70 * Math.Cos(3 * t) - 40 * Math.Sin(3 * t)) / 10;

Some text

\(\quad\)

Left figure: Trefoil A. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Trefoil B. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 0^\circ\).

Some text

\(\quad\)

Left figure: Trefoil A. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Trefoil B. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 0^\circ\).

Cinquefoil Knot

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

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

Cinquefoil Knot: https://home.adelphi.edu/~stemkoski/knotgallery/

An example in C#

var x = Math.Cos(2 * t) * (3 + Math.Cos(5 * t)) / 10;
var y = Math.Sin(2 * t) * (3 + Math.Cos(5 * t)) / 10;
var z = Math.Sin(5 * t) / 10;

Some text

\(\quad\)

Left figure: Cinquefoil Knot. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Cinquefoil Knot. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 0^\circ\).

Some text

\(\quad\)

Left figure: Cinquefoil Knot. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Cinquefoil Knot. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 0^\circ\).

74 Knot

See also: https://en.wikipedia.org/wiki/74_knot

This is a Lissajous knot.

See also: https://katlas.org/wiki/7_4

An example in C#

var x = Math.Cos(2 * t + 0.22) / 10;
var y = Math.Cos(3 * t + 1.10) / 10;
var z = Math.Cos(7 * t) / 10;

Some text

\(\quad\)

Left figure: 74 Knot. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: 74 Knot. Perspective camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Some text

\(\quad\)

Left figure: 74 Knot. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: 74 Knot. Orthographic camera. Camera angles are \(\theta=180^\circ\) and \(\phi = 90^\circ\).

Figure-eight knot

See also: https://mathcurve.com/courbes3d.gb/noeuds/noeudenhuit.shtml

See also: https://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)

See also: https://katlas.org/wiki/4_1

An example in C#

var x = ((2 + Math.Cos(2 * t)) * Math.Cos(3 * t)) / 10;
var y = ((2 + Math.Cos(2 * t)) * Math.Sin(3 * t)) / 10;
var z = (Math.Sin(4 * t)) / 10;

Some text

\(\quad\)

Left figure: Figure-eight_knot. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Figure-eight_knot. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 180^\circ\).

Some text

Some text

\(\quad\)

Left figure: Figure-eight_knot. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Figure-eight_knot. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 180^\circ\).