Polynomial knots

Polynomial knots#

See also: https://home.adelphi.edu/~stemkoski/knotgallery/

Polynomial knot 3-1#

Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
double t5 = t4 * t;
var x = (t3 - 3 * t) / 10;
var y = (t4 - 4 * t2) / 10;
var z = (t5 / 5 - 2 * t) / 10;

PathP_Poly31Knot_a \(\quad\) PathO_Poly31Knot_a

Left figure: Polynomial knot 3-1. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Polynomial knot 3-1. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Polynomial knot 4-1#

Figure-Eight (41) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t4 = t2 * t2;
var x = (2.0 / 5.0) * t * (t2 - 7) * (t2 - 10) / 10;
var y = (t4 - 13 * t2) / 10;
var z = (1.0 / 10.0) * t * (t2 - 4) * (t2 - 9) * (t2 - 12) / 10;

PathP_Poly41Knot_a \(\quad\) PathO_Poly41Knot_a

Left figure: Polynomial knot 4-1. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Polynomial knot 4-1. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Polynomial knot 5-1#

Cinquefoil (51) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
double t5 = t4 * t;
double t6 = t5 * t;
double t7 = t6 * t;
var x = (1.0 / 5.0) * (t5 - 36 * t3 + 260 * t) / 10;
var y = (1.0 / 2.0) * (t4 - 24 * t2) / 10;
var z = (1.0 / 100.0) * (t7 - 31 * t5 + 164 * t3 - 560 * t) / 10;

PathP_Poly51Knot_a \(\quad\) PathO_Poly51Knot_a

Left figure: Polynomial knot 5-1. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Polynomial knot 5-1. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Polynomial knot 6-2#

Six-Crossing (62) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
var x = (3.0 / 4.0) * t * (t2 - 4) * (t2 - 11) / 10;
var y = (t4 - 12 * t2) / 10;
var z = (1.0 / 200.0) * t * (t2 - 1) * (t2 - 3) * (t2 - 9) * (t2 - 10) * (t2 - 12) / 10;

PathP_Poly62Knot_a \(\quad\) PathO_Poly62Knot_a

Left figure: Polynomial knot 6-2. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Polynomial knot 6-2. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Polynomial knot 7-4#

Seven-Crossing (74) Polynomial Knot. https://home.adelphi.edu/~stemkoski/knotgallery/poly3.html

double t2 = t * t;
double t3 = t2 * t;
double t4 = t3 * t;
double t5 = t4 * t;
var x = (4.0 / 5.0) * t * (t2 - 6) * (t2 - 12) / 10;
var y = t2 * (t2 - 7) * (t2 - 9) / 10;
var z = (1.0 / 200.0) * t * (t2 - 0.2) * (t2 - 1) * (t2 - 5) * (t2 - 6.5) * (t2 - 9) * (t2 - 10) / 10;

PathP_Poly74Knot_a \(\quad\) PathO_Poly74Knot_a

Left figure: Polynomial knot 7-4. Perspective camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

Right figure: Polynomial knot 7-4. Orthographic camera. Camera angles are \(\theta=90^\circ\) and \(\phi = 90^\circ\).

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