quatConj: Quaternion Conjugate Operation

Quaternion Conjugate Calculation

The quatConj function computes the quaternion conjugate by negating the vector part while preserving the scalar component. For a unit quaternion, the conjugate represents the inverse rotation.

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

$$q^* = (-x, -y, -z, w)$$

Key Properties:

• For unit quaternions: $q \cdot q^* = 1$ (identity quaternion)
• The conjugate represents the inverse rotation: $q^{-1} = q^*$ when $|q| = 1$
• Conjugation reverses rotation direction while preserving rotation axis

This operation is used for transforming vectors back to original coordinate systems and implementing geometric reflection algorithms.

Symmetric Reflection Patterns

This example demonstrates how quaternion conjugates create symmetric reflection patterns through mirror transformations.

ライブエディター

const fragment = () => {
      const center = vec2(0.5, 0.5)
      const radius = uv.sub(center).length()
      const angle = atan2(uv.y.sub(0.5), uv.x.sub(0.5))

      const q = vec4(
              angle.add(iTime).sin(),
              angle.mul(2).cos(),
              radius.mul(5).add(iTime.mul(0.7)).sin(),
              radius.mul(3).sub(iTime.mul(0.5)).cos()
      ).normalize()

      const conjugate = quatConj(q)
      const symmetryField = dot(q, conjugate)

      const originalRotation = quat2mat3(q).mul(vec3(uv.sub(0.5), 0))
      const mirrorRotation = quat2mat3(conjugate).mul(vec3(uv.sub(0.5), 0))

      const pattern1 = originalRotation.xy.mul(8).sin().length()
      const pattern2 = mirrorRotation.xy.mul(8).cos().length()
      const symmetry = pattern1.add(pattern2).mul(symmetryField.abs())

      const color = vec3(symmetry.mul(0.7), symmetry, symmetry.mul(0.5))
      return vec4(color, 1)

}

The quatConj function provides quaternion conjugation operations for inverse rotations and reflection operations in 3D space.