quatNeg: Quaternion Negation Operations

Double Cover Mapping and Orientation Reversal

The quatNeg function implements quaternion negation, creating rotational equivalence through the double cover property of quaternions. For any quaternion $q = (x, y, z, w)$, its negation $-q = (-x, -y, -z, -w)$ represents the same geometric rotation, demonstrating that each 3D rotation has exactly two quaternion representations.

Mathematical Definition:

For quaternion $q = (x, y, z, w)$:

$$\text{quatNeg}(q) = (-x, -y, -z, -w)$$

Double Cover Property:

• Both $q$ and $-q$ represent identical rotations in 3D space
• Negation preserves rotation magnitude and axis direction
• Creates opposite signs while maintaining rotational equivalence
• Essential for interpolation normalization and angle minimization

ライブエディター

const fragment = () => {
      const cellPos = uv.mul(8)
      const cell = cellPos.floor()
      const local = cellPos.fract().sub(0.5)

      const hash = cell.x.mul(127.1).add(cell.y.mul(311.7)).sin().mul(43758.5).fract()
      const angle = hash.mul(6.28).add(iTime.mul(0.5))

      const q = vec4(angle.sin().mul(0.7), angle.mul(1.3).sin().mul(0.5), angle.mul(0.8).sin(), angle.cos()).normalize()
      const negQ = quatNeg(q)

      const useOriginal = hash.greaterThan(0.5)
      const selected = q.select(negQ, useOriginal)

      const rotMatrix = quat2mat3(selected)
      const rotated = rotMatrix.mul(vec3(local, hash.sub(0.5)))

      const pattern = rotated.x.mul(rotated.y).mul(8).sin()
      const intensity = pattern.abs()
      const color = vec3(intensity, intensity.mul(0.7), intensity.mul(0.4))

      return vec4(color, 1)

}