Quaternion Inverse

Compute the inverse quaternion for reversing rotations

Quaternion Inverse Calculator

Quaternion Inverse

Computes the inverse quaternion q⁻¹ of a given quaternion, used for reversing rotations and coordinate transformations

Inverse quaternion properties

Formula: q⁻¹ = q* / |q|² (conjugate divided by norm squared)
Property: q · q⁻¹ = 1 (identity quaternion)
Use: Reversing rotations and transformations

Enter quaternion for inverse calculation
Input quaternion (q)
Settings
Note: Division by zero is not possible. Ensure |q| ≠ 0.
Quaternion inverse result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Inverse: q⁻¹ = q* / |q|² with property q · q⁻¹ = 1

Inverse Info

Inverse properties

Formula: q⁻¹ = q* / |q|²
Identity: q · q⁻¹ = 1
For unit q: q⁻¹ = q*

Rotation reversal Transformation q · q⁻¹ = 1

Warning: Defined only for q ≠ 0
Tip: For unit quaternions: q⁻¹ = q*

Calculation steps
1. Form conjugate: q* = w - xi - yj - zk
2. Norm squared: |q|² = w² + x² + y² + z²
3. Division: q⁻¹ = q* / |q|²
4. Verification: q · q⁻¹ = 1

Formulas for the Quaternion Inverse

General inverse formula
\[q^{-1} = \frac{q^*}{|q|^2} = \frac{w - xi - yj - zk}{w^2 + x^2 + y^2 + z^2}\]

Conjugate divided by the norm squared

Component-wise calculation
\[\begin{align} w_{inv} &= \frac{w}{|q|^2} \\ x_{inv} &= \frac{-x}{|q|^2} \\ y_{inv} &= \frac{-y}{|q|^2} \\ z_{inv} &= \frac{-z}{|q|^2} \end{align}\]

Each component computed individually

Special case: unit quaternion
\[\text{If } |q| = 1: \quad q^{-1} = q^*\]

For normalized quaternions the inverse equals the conjugate

Verification property
\[q \cdot q^{-1} = q^{-1} \cdot q = 1\]

Quaternion multiplied by its inverse yields the identity

Norm squared calculation
\[|q|^2 = q \cdot q^* = w^2 + x^2 + y^2 + z^2\]

Sum of squares of all components

Examples for Quaternion Inverse

Example 1: General quaternion
q = 1 + 3i + 5j + 2k
Step 1: Conjugate \[q^* = 1 - 3i - 5j - 2k\] Step 2: Norm squared \[|q|^2 = 1^2 + 3^2 + 5^2 + 2^2 = 1 + 9 + 25 + 4 = 39\] Step 3: Inverse \[q^{-1} = \frac{1 - 3i - 5j - 2k}{39}\]

q⁻¹ = 1/39 - 3/39i - 5/39j - 2/39k

Example 2: Unit quaternion
q = 0.5 + 0.5i + 0.5j + 0.5k |q| = 1
Check unit length: \[|q|^2 = 0.5^2 + 0.5^2 + 0.5^2 + 0.5^2 = 1\] For unit quaternions: \[q^{-1} = q^* = 0.5 - 0.5i - 0.5j - 0.5k\] Verification: \[q \cdot q^{-1} = 1\]

Unit q: Inverse = Conjugate

Example 3: Verification
q = 2 + 1i + 0j + 0k
Compute inverse: \[|q|^2 = 4 + 1 + 0 + 0 = 5\] \[q^{-1} = \frac{2 - 1i}{5} = 0.4 - 0.2i\] Verification: \[q \cdot q^{-1} = (2 + i)(0.4 - 0.2i) = 1 + 0i\]

✓ Verification successful

Example 4: Zero quaternion
q = 0 + 0i + 0j + 0k
Norm squared: \[|q|^2 = 0^2 + 0^2 + 0^2 + 0^2 = 0\] Problem: \[q^{-1} = \frac{q^*}{0} = \text{UNDEFINED}\] Division by zero!

Inverse does not exist

Practical applications of the inverse
Rotation reversal
Undo animation
Coordinate transformation
System change
Inverse kinematics
Robot control
Calibration
Sensor correction

The inverse is essential for reversing rotations and transformations in 3D applications

Applications of the Quaternion Inverse

The quaternion inverse is fundamental for many 3D applications:

3D Graphics & Animation
  • Rotation reversal: undo animation
  • Coordinate transformation: Between coordinate systems
  • Inverse transformations: undo functionality
  • Camera control: reverse viewing direction
Robotics & Kinematics
  • Inverse kinematics: target position → joint angles
  • Path planning: backward movements
  • Calibration: sensor offset correction
  • Orientation control: desired vs actual deviation
Aerospace
  • Attitude control: orientation correction
  • Navigation: reference frame transformations
  • Stabilization: disturbance compensation
  • Maneuver planning: reverse moves
Key properties
  • Existence: Defined only for q ≠ 0
  • Unit quaternions: q⁻¹ = q* (simpler)
  • Verification: check q · q⁻¹ = 1
  • Numerical stability: be careful with small |q|

Quaternion Inverse: the key to rotation reversal

The quaternion inverse is a fundamental operation for reversing 3D rotations and transformations. It is based on the mathematical definition q⁻¹ = q*/|q|², where q* is the conjugate and |q|² the norm squared. For unit quaternions, commonly used in 3D graphics, the calculation simplifies to q⁻¹ = q*, because |q| = 1. The inverse allows exact reversal of rotations, coordinate system transformations, and solving inverse kinematics problems. It is essential for modern 3D engines, robotics applications and simulations.

Summary

The quaternion inverse is more than just a mathematical operation - it is the tool for precise rotation reversal in 3D applications. Its special importance lies in the property q · q⁻¹ = 1, making it the perfect undo operation. For unit quaternions the computation is particularly elegant, since the inverse equals the conjugate. This efficiency makes it suitable for real-time applications in gaming, VR/AR and robotics where millions of calculations per second are required. Understanding the inverse is therefore indispensable for anyone working with 3D rotations.

More Quaternion Functions

Addition  •  Subtraction  •  Division  •  Multiplication  •  Concatenate  •  Length  •  Interpolation  •  Normalize  •  Scalar Multiplication  •  Dot Product  •  Yaw-Pitch-Roll  •  Conjugates  •  Inverse  •  Negation  •