rotate3dY: Specialized Y-Axis Rotation Matrix

Mathematical Foundation of Y-Axis Rotation

The rotate3dY function constructs a specialized 3×3 rotation matrix for rotations around the Y-axis. This optimized implementation provides computational efficiency for vertical axis rotations commonly used in navigation and camera systems.

The Y-axis rotation matrix follows the standard mathematical form:

$$R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}$$

Geometric Properties and Spatial Behavior

Y-axis rotation preserves the Y-coordinate while transforming X and Z coordinates through circular motion in the XZ-plane:

Axis Preservation: Vectors along the Y-axis $(0,1,0)$ remain unchanged under this transformation.

XZ-Plane Rotation: The transformation rotates vectors within the XZ-plane, creating horizontal circular motion.

Navigation Systems: Fundamental operation for yaw rotations in 3D navigation and camera control systems.

Vortex Pattern Formation

This demonstration explores Y-axis rotation applied to create vortex-like patterns, showcasing the mathematical beauty of rotational flow fields in computational 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(2)
  const distance = pos.length()
  const angle = distance.mul(4).add(iTime.mul(1.5))
  const rotated = rotate3dY(angle).mul(pos)
  const vortex = rotated.x.mul(6).sin().mul(rotated.z.mul(6).cos())
  const intensity = vortex.mul(smoothstep(1.5, 0, distance))
  const color = vec3(intensity.add(0.3), intensity.mul(0.7).add(0.2), intensity.mul(0.9).add(0.1))
  return vec4(color, 1)
}