rotate4dZ: Four-Dimensional Z-Axis Planar Rotation

Homogeneous Planar Transformation Theory

The rotate4dZ function generates a 4×4 homogeneous rotation matrix for rotations around the Z-axis, extending 2D planar rotation into projective coordinate space. This transformation preserves Z and W coordinates while rotating vectors in the XY-plane.

The mathematical structure follows:

$$R_{4D,z}(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Mathematical Properties and Applications

This transformation bridges 2D rotation mathematics with 4D homogeneous coordinate systems:

Complex Plane Isomorphism: XY-plane rotation corresponds to complex multiplication $z \mapsto z \cdot e^{i\theta}$.

Screen Space Operations: Direct application for 2D UI transformations within 3D rendering pipelines.

Projective Invariance: Maintains consistency with perspective projection and camera transformations.

Plasma Confinement Dynamics

This demonstration models plasma confinement in tokamak reactors, visualizing magnetic field containment through Z-axis rotation that creates toroidal plasma flows essential for fusion energy.

ライブエディター

const fragment = () => {
  const center = vec3(0.5, 0.5, 0)
  const pos = vec3(uv, 0).sub(center).mul(4)
  const toroidal = pos.length().sub(1.2)
  const fieldAngle = pos.y.atan2(pos.x).mul(4).add(iTime.mul(3))
  const rotation = rotate4dZ(fieldAngle)
  const confined = rotation.mul(vec4(pos, 1))
  const plasma = toroidal.abs().negate().mul(8).exp()
  const temperature = confined.x.mul(6).sin().mul(confined.y.mul(6).cos())
  const fusion = plasma.mul(temperature.add(1)).mul(0.5)
  const heat = fusion.pow(0.4)
  const color = vec3(heat, heat.mul(0.7).add(0.3), heat.mul(0.4).add(0.6))
  return vec4(color, 1)
}