rotate4dX: Specialized Four-Dimensional X-Axis Rotation

Homogeneous X-Axis Transformation Theory

The rotate4dX function generates a specialized 4×4 homogeneous rotation matrix for rotations around the X-axis. This transformation preserves the X-coordinate and W-coordinate while rotating vectors in the YZ-plane within the projective coordinate framework.

The 4×4 X-axis rotation matrix follows the canonical form:

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

Projective Geometry and Dimensional Folding

X-axis rotation in homogeneous coordinates exhibits unique mathematical properties that bridge 3D rotation with projective transformations:

Projective Invariance: The transformation maintains projective ratios while enabling seamless pipeline integration.

Dimensional Embedding: 3D rotation is embedded within 4D space, enabling composition with translation and scaling operations.

Perspective Compatibility: Direct integration with perspective projection matrices for camera transformations.

Magnetic Field Line Topology

This example demonstrates magnetic field line behavior under rotational perturbations, visualizing how electromagnetic field topology evolves through continuous X-axis rotations in four-dimensional parameter space.

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(5)
  const fieldAngle = pos.length().mul(0.8).add(iTime.mul(0.4))
  const rotation = rotate4dX(fieldAngle)
  const fieldVector = rotation.mul(vec4(pos, 1))
  const fieldLines = fieldVector.y.mul(2).sin().mul(fieldVector.z.mul(2).cos())
  const magnetic = fieldLines.abs().pow(0.7)
  const topology = smoothstep(0.3, 0.8, magnetic)
  const flux = fieldVector.y.mul(fieldVector.z).mul(0.01).add(0.5)
  const color = vec3(topology.mul(flux), topology.mul(0.6), topology.mul(0.9))
  return vec4(color, 1)
}

結果

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