rotate4dY: Specialized Four-Dimensional Y-Axis Rotation

Homogeneous Y-Axis Transformation Theory

The rotate4dY function generates a specialized 4×4 homogeneous rotation matrix for rotations around the Y-axis in projective coordinate space. This transformation preserves the Y-coordinate and homogeneous W-coordinate while rotating vectors in the XZ-plane.

The 4×4 Y-axis rotation matrix follows the mathematical form:

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

Dimensional Navigation and Spatial Topology

Y-axis rotation in homogeneous space exhibits profound mathematical connections to navigation systems and spatial topology:

Yaw Navigation: Fundamental operation for horizontal orientation changes in 3D navigation systems.

Topological Preservation: Maintains spatial relationships while enabling continuous orientation changes.

Projective Stability: Ensures stable behavior under perspective transformations and camera operations.

Molecular Orbital Electron Dynamics

This example visualizes quantum mechanical electron orbitals under rotational symmetry, demonstrating how Y-axis rotation reveals the mathematical structure of atomic electron probability distributions in four-dimensional phase space.

ライブエディター

const fragment = () => {
  const center = vec3(0.5, 0.5, 0)
  const pos = vec3(uv, iTime.mul(0.3).sin().mul(0.2)).sub(center).mul(6)
  const orbitalAngle = pos.length().mul(1.5).add(iTime.mul(0.5))
  const rotation = rotate4dY(orbitalAngle)
  const orbital = rotation.mul(vec4(pos, 1))
  const radial = orbital.length()
  const angular = orbital.z.atan2(orbital.x).mul(3)
  const probability = radial.negate().exp().mul(angular.sin().pow(2))
  const electron = probability.mul(smoothstep(0.1, 0.3, probability))
  const energy = radial.mul(0.1).add(0.5)
  const color = vec3(electron.mul(0.4), electron.mul(energy), electron)
  return vec4(color, 1)
}