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A torus knot is a closed curve that winds along the surface of the torus in three dimensions, which can be defined by two parameters p and q, which represent the number of times the curve winds around the major and minor circles of the torus surface, respectively

The mathematical description of a ring surface knot can be expressed by the following parametric equations:

x(u,v)=(R+rcos⁡(qv))cos⁡(pv)y(u,v)=(R+rcos⁡(qv))sin⁡(pv)z(u,v)=rsin⁡(qv)\begin{align*} x(u, v) &= (R + r \cos(qv)) \cos(pv) \\ y(u, v) &= (R + r \cos(qv)) \sin(pv) \\ z(u, v) &= r \sin(qv) \end{align*}

Vector normalization involves scaling the length of a vector to 1, keeping its direction constant. For any non-zero vectorv⃗=(x,y,z)\vec{v} = (x, y, z),its normalized vectorv⃗norm\vec{v}_{norm}The formula for this is:

v⃗norm=v⃗∣v⃗∣=(xx2+y2+z2,yx2+y2+z2,zx2+y2+z2)\vec{v}_{norm} = \frac{\vec{v}}{|\vec{v}|} = \left( \frac{x}{\sqrt{x^2 + y^2 + z^2}}, \frac{y}{\sqrt{x^2 + y^2 + z^2}}, \frac{z}{\sqrt{x^2 + y^2 + z^2}} \right)

In three-dimensional space, rotation about any axis can be realized by means of a rotation matrix. For example, a rotation of one angle around the x-axisθ\thetaThe rotation matrix is:

Rx(θ)=(1000cos⁡(θ)−sin⁡(θ)0sin⁡(θ)cos⁡(θ))R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{pmatrix}