How do you invert a quaternion?

The inverse of a non-zero quaternion q is calculated by dividing its conjugate q' by the product of q and its conjugate, q*q'.

The inverse of a quaternion refers to the multiplicative inverse (or 1/q), and as described in the reference, it can be computed using a specific formula for any non-zero quaternion.

The Formula for Quaternion Inverse

For any non-zero quaternion q, its inverse, denoted q⁻¹, is computed by the following formula:

*`q⁻¹ = q' / (qq')`**

This formula provides the multiplicative inverse, meaning that q * q⁻¹ = q⁻¹ * q = 1, where 1 is the identity quaternion (1 + 0i + 0j + 0k).

Understanding the Components

To use the inversion formula effectively, you need to understand its parts:

• Quaternion (q): A quaternion is typically written in the form q = w + xi + yj + zk, where w is the scalar part and xi + yj + zk is the vector part. w, x, y, and z are real numbers.
• Conjugate of a Quaternion (q'): The conjugate of q is found by negating the vector part while keeping the scalar part the same. If q = w + xi + yj + zk, its conjugate is q' = w - xi - yj - zk.
• Product of a Quaternion and its Conjugate (q*q'): Multiplying a quaternion by its conjugate yields a scalar value equal to the sum of the squares of its components: q*q' = (w + xi + yj + zk) * (w - xi - yj - zk) = w² + x² + y² + z². This scalar is also the squared magnitude (or squared norm) of the quaternion, often written as |q|². Since q is non-zero, |q|² will be a positive scalar.

Therefore, the inversion formula can also be expressed as q⁻¹ = q' / |q|².

Step-by-Step Inversion

To invert a non-zero quaternion q = w + xi + yj + zk:

1. Calculate the conjugate: Find q' = w - xi - yj - zk.
2. Calculate the squared magnitude: Compute |q|² = w² + x² + y² + z². This is the same as calculating q*q'.
3. Divide: Divide each component of the conjugate q' by the scalar |q|²:
   q⁻¹ = (w / |q|²) - (x / |q|²)i - (y / |q|²)j - (z / |q|²)k.

Example Calculation

Let's invert the quaternion q = 2 + i - 3j + 4k.

1. Conjugate (q'): q' = 2 - i + 3j - 4k.
2. Squared Magnitude (|q|² or q*q'):
   |q|² = 2² + 1² + (-3)² + 4²
|q|² = 4 + 1 + 9 + 16
|q|² = 30.
   (Confirming q*q' = 30 as well).
3. Inverse (q⁻¹): Divide q' by |q|²:
   q⁻¹ = (2 - i + 3j - 4k) / 30
q⁻¹ = (2/30) - (1/30)i + (3/30)j - (4/30)k
q⁻¹ = (1/15) - (1/30)i + (1/10)j - (2/15)k

The inverse of 2 + i - 3j + 4k is (1/15) - (1/30)i + (1/10)j - (2/15)k.

Summary Table

This table illustrates the terms and values involved in the inversion process for the example quaternion.

Importance

Quaternion inversion is crucial because it allows you to 'undo' a rotation or transformation represented by a quaternion. For instance, if q rotates an object from orientation A to B, then q⁻¹ rotates it from B back to A.