quatInverse: Quaternion Inverse Calculation

Mathematical Inverse Operation

The quatInverse function calculates the true mathematical inverse of a quaternion using the formula $q^{-1} = \frac{q^*}{||q||^2}$. This operation produces the exact rotational inverse, undoing the original rotation when composed.

Mathematical Definition: For quaternion $q$:

$$q^{-1} = \frac{q^*}{||q||^2} = \frac{\text{quatConj}(q)}{\text{quatLengthSq}(q)}$$

Properties:

• For any quaternion: $q \cdot q^{-1} = q^{-1} \cdot q = (0,0,0,1)$ (identity)
• Reverses the rotation represented by the quaternion
• Handles non-unit quaternions correctly through magnitude normalization

This function is essential for undoing rotations and solving rotational equations.

Rotational Error Analysis

This example demonstrates quaternion inverse accuracy by applying a rotation followed by its inverse and measuring the restoration error.

ライブエディター

const fragment = () => {
      const cellPos = uv.mul(6)
      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 baseQ = vec4(hash.mul(6.28).sin(), hash.mul(3.14).cos(), hash.mul(9.42).sin(), hash.mul(1.57).cos())

      const scaledQ = baseQ.mul(hash.mul(2).add(1))
      const invQ = quatInverse(scaledQ)

      const testVector = vec3(local.x, local.y, hash)
      const rotated = quat2mat3(scaledQ).mul(testVector)
      const recovered = quat2mat3(invQ).mul(rotated)

      const errorMagnitude = recovered.sub(testVector).length()
      const precision = errorMagnitude.add(0.1).reciprocal()

      const visualField = precision.mul(0.1).sin().abs()
      const color = vec3(visualField, visualField.mul(0.7), visualField.mul(0.3))

      return vec4(color, 1)

}

The quatInverse function provides accurate quaternion inversion for precise rotational calculations and error correction in rotation sequences.