Quaternion Transformations

Conjugate, inverse, normalization and other quaternion operations

Quaternion Transformations Calculator

Quaternion Transformations

Computes various transformations of a quaternion: conjugate, inverse, normalization and negation

Available transformations

Conjugate: Negate vector parts
Inverse: For rotation reversal
Normalization: Make unit length
Negation: Flip all signs

Enter quaternion for transformation
Input quaternion (q)
Settings
Quaternion transformation result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Transformation: f(q) → q' (Conjugate, Inverse, Normalize or Negate)

Transformations Info

Transformations

Conjugate: Negate vector parts
Inverse: For rotation reversal
Normalization: Unit length
Negation: Flip all signs

Conjugate Inverse Normalization

Commonly used: Conjugate and Inverse
Important: Normalize for unit quaternions

Quick reference
q*: Conjugate (w, -x, -y, -z)
q⁻¹: Inverse = q*/|q|²
q̂: Normalized = q/|q|
-q: Negated (-w, -x, -y, -z)

Formulas for Quaternion Transformations

Conjugate
\[q^* = w - xi - yj - zk\]

Negate the vector components

Inverse
\[q^{-1} = \frac{q^*}{|q|^2} = \frac{q^*}{w^2 + x^2 + y^2 + z^2}\]

Conjugate divided by norm squared

Normalization
\[\hat{q} = \frac{q}{|q|} = \frac{w + xi + yj + zk}{\sqrt{w^2 + x^2 + y^2 + z^2}}\]

Divide by the magnitude to make unit length

Negation
\[-q = -w - xi - yj - zk\]

Flip the signs of all components

Important properties
\[\begin{align} (q^*)^* &= q \\ |q^*| &= |q| \\ q \cdot q^{-1} &= 1 \\ |\hat{q}| &= 1 \end{align}\]

Fundamental transformation properties

Examples for Quaternion Transformations

Example 1: Conjugate
q = 1 + 3i + 5j + 2k
Conjugate calculation: \[\begin{align} q^* &= w - xi - yj - zk \\ &= 1 - 3i - 5j - 2k \end{align}\] Only vector parts negated!

q* = 1 - 3i - 5j - 2k

Example 2: Normalization
q = 1 + 3i + 5j + 2k
Step 1: Compute magnitude \[|q| = \sqrt{1^2 + 3^2 + 5^2 + 2^2} = \sqrt{39} ≈ 6.245\] Step 2: Normalize \[\hat{q} = \frac{q}{|q|} = \frac{1 + 3i + 5j + 2k}{6.245}\]

|q̂| = 1 (unit length)

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

q·q⁻¹ = 1 (identity)

Example 4: Negation
q = 1 + 3i + 5j + 2k
Negation (all components): \[\begin{align} -q &= -w - xi - yj - zk \\ &= -1 - 3i - 5j - 2k \end{align}\] Same rotation, opposite direction

-q = -1 - 3i - 5j - 2k

Practical applications
Conjugate
Rotation reversal
Inverse
Coordinate transformation
Normalization
Unit quaternion
Negation
Alternative representation

These transformations are fundamental for rotation computations in 3D applications

Step-by-step guide
Conjugate & Negation
  1. Enter quaternion: w + xi + yj + zk
  2. Conjugate: w - xi - yj - zk
  3. Negation: -w - xi - yj - zk
  4. Apply result directly
Inverse & Normalization
  1. Compute magnitude: √(w²+x²+y²+z²)
  2. Normalize: q/|q|
  3. Inverse: q*/|q|² (if |q| ≠ 0)
  4. Verify: q·q⁻¹ = 1

Applications of Quaternion Transformations

The quaternion transformations are essential for many 3D applications:

3D Graphics & Animation
  • Conjugate: Rotation reversal for animations
  • Inverse: Coordinate system transformations
  • Normalization: Clean unit quaternions
  • Negation: Alternative rotation representation
Robotics & Control
  • Inverse: Inverse kinematics
  • Conjugate: Sensor calibration
  • Normalization: Drift correction
  • Transformations: Coordinate conversion
Aerospace
  • Attitude control: orientation correction
  • Navigation: reference frame transformations
  • Stabilization: disturbance compensation
  • Maneuver planning: reverse moves
VR/AR & Motion Capture
  • Head tracking: orientation correction
  • Hand tracking: joint transformations
  • Calibration: coordinate alignment
  • Drift correction: long-term stability

Quaternion Transformations: Building blocks of 3D rotation

Quaternion transformations form the foundation for efficient 3D rotation computations. The conjugate enables inverting rotations and is essential for coordinate transformations. The inverse is fundamental for undoing rotations, while normalization ensures quaternions maintain unit length and thus represent valid rotations. Negation provides an alternative representation of the same rotation. These operations are highly optimized and numerically stable, making them ideal for real-time applications in gaming, robotics and VR/AR.

Summary

The four fundamental quaternion transformations - conjugate, inverse, normalization and negation - are indispensable tools for any 3D application. They allow manipulating, reversing and correcting rotations. It is important to understand that the conjugate reverses rotation, the inverse is the mathematical undo, normalization provides numerical stability, and negation yields an equivalent rotation representation. Modern 3D engines use these operations millions of times per second for realistic animations and precise rotation computations.

More Quaternion Functions

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