Quaternion Concatenation

Concatenation of rotations via quaternion multiplication

Quaternion Concatenation Calculator

Quaternion Concatenation (Concatenation)

Concatenates two rotation quaternions q₁ ∘ q₂ by quaternion multiplication for sequential application of rotations

Concatenation Properties

Operation: Concatenation = quaternion multiplication (Hamilton product)
Order: q₁ ∘ q₂ means apply q₁ first, then q₂
Use: Sequential rotations, coordinate transforms

Enter two quaternions to concatenate
First Rotation (q₁)
Second Rotation (q₂)
q₁ ∘ q₂: Apply q₁ first, then q₂
Concatenation ≡ Multiplication: Mathematically identical
Quaternion Concatenation Result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Concatenation: q₁ ∘ q₂ = q₁ × q₂ (Hamilton product for sequential rotations)

Concatenation Info

Concatenation Properties

Identical: Concatenation = multiplication
Sequential: Apply q₁, then q₂
Not commutative: q₁∘q₂ ≠ q₂∘q₁

Sequential Not commutative Rotation chain

Order: q₁ ∘ q₂ ≠ q₂ ∘ q₁
Identity: Concatenation = multiplication

Main applications
Animation: Motion sequences
Robotics: Joint rotations
3D Graphics: Object transforms
Navigation: Course corrections


Formulas for Quaternion Concatenation

Concatenation formula (Hamilton product)
\[q_1 \circ q_2 = q_1 \times q_2 = (w_1 + x_1 i + y_1 j + z_1 k) \times (w_2 + x_2 i + y_2 j + z_2 k)\]

Concatenation is mathematically identical to quaternion multiplication

Component-wise calculation
\[\begin{align} w &= w_1 w_2 - x_1 x_2 - y_1 y_2 - z_1 z_2 \\ x &= w_1 x_2 + x_1 w_2 + y_1 z_2 - z_1 y_2 \\ y &= w_1 y_2 - x_1 z_2 + y_1 w_2 + z_1 x_2 \\ z &= w_1 z_2 + x_1 y_2 - y_1 x_2 + z_1 w_2 \end{align}\]

Full Hamilton product formula

Rotation interpretation
\[\text{Object} \xrightarrow{q_1} \text{Intermediate} \xrightarrow{q_2} \text{Final}\] \[\text{Total rotation} = q_1 \circ q_2\]

Sequential application of rotations

Associativity property
\[(q_1 \circ q_2) \circ q_3 = q_1 \circ (q_2 \circ q_3)\]

Grouping does not matter

Non-commutativity
\[q_1 \circ q_2 \neq q_2 \circ q_1\] \[\text{Order matters!}\]

Different orders produce different results

Examples for Quaternion Concatenation

Example 1: Rotation concatenation
q₁ = 3 + 2i + 4j + 1k (first rotation) q₂ = 1 + 3i + 5j + 2k (second rotation)
Concatenation q₁ ∘ q₂: \[\begin{align} w &= 3·1 - 2·3 - 4·5 - 1·2 = 3-6-20-2 = -25 \\ x &= 3·3 + 2·1 + 4·2 - 1·5 = 9+2+8-5 = 14 \\ y &= 3·5 - 2·2 + 4·1 + 1·3 = 15-4+4+3 = 18 \\ z &= 3·2 + 2·5 - 4·3 + 1·1 = 6+10-12+1 = 5 \end{align}\]

q₁ ∘ q₂ = -25 + 14i + 18j + 5k

Example 2: Order matters!
q₁ = 1 + i + 0j + 0k (90° about X) q₂ = 1 + 0i + j + 0k (90° about Y)
q₁ ∘ q₂ (X then Y): \[= 1 + i + j + k\] q₂ ∘ q₁ (Y then X): \[= 1 + i + j - k\] Different results!

q₁∘q₂ ≠ q₂∘q₁

Example 3: Robot arm motion
Shoulder rotation q₁ Elbow rotation q₂
Total arm orientation: \[\text{Endeffector} = q_{\text{shoulder}} \circ q_{\text{elbow}}\] Motion sequence: \[\text{Base} \xrightarrow{q_1} \text{Upper} \xrightarrow{q_2} \text{Hand}\]

Concatenation = joint hierarchy

Example 4: Identity concatenation
q = 2 + 3i + 1j + 4k e = 1 + 0i + 0j + 0k (identity)
Concatenation with identity: \[\begin{align} q \circ e &= q \\ e \circ q &= q \end{align}\] Identity changes nothing

Identity is neutral

Practical applications of concatenation
Animation
Motion sequences
Robotics
Joint chains
3D Graphics
Object hierarchies
Navigation
Course changes

Concatenation enables building complex rotation sequences from simple single rotations

Step-by-step guide
Conceptual
  1. Define first rotation q₁
  2. Define second rotation q₂
  3. Decide order: q₁ ∘ q₂
  4. Understand concatenation = multiplication
Computation
  1. Apply Hamilton product formula
  2. Compute all 16 terms
  3. Combine into four components
  4. Interpret result as new rotation

Applications of Quaternion Concatenation

Quaternion concatenation is fundamental for complex 3D rotation sequences:

3D Graphics & Animation
  • Object hierarchies: parent-child transforms
  • Skeletal animation: bone chains
  • Camera motion: complex paths
  • Particle systems: rotation inheritance
Robotics & Automation
  • Joint chains: serial kinematics
  • Robot arms: end-effector orientation
  • Path planning: sequential motions
  • Coordinate transforms: between systems
Aerospace
  • Maneuver sequences: multiple rotations
  • Attitude control: stabilization chains
  • Navigation: course corrections
  • Docking operations: precision moves
Important properties
  • Not commutative: order matters
  • Associative: grouping arbitrary
  • Identical: Concatenation = multiplication
  • Efficient: optimized for realtime

Quaternion Concatenation: mastering sequential rotations

Quaternion concatenation is mathematically identical to quaternion multiplication, but conceptually focused on the sequential application of rotations. It allows composing complex motions from simple single rotations. The non-commutative nature of the operation reflects the physical reality that rotation order affects the final outcome. In practice, concatenation is used for hierarchies of objects such as robotic links, skeletal animations or coordinate system transforms. Associativity allows efficient precomputation of sub-chains.

Summary

Quaternion concatenation is the operation of choice when sequential rotations are required. Its mathematical elegance — based on the Hamilton product — combines with practical versatility for complex 3D systems. From simple object hierarchies in games to precise robotic controls, concatenation enables intuitive modeling of motion sequences. Understanding non-commutativity is crucial for correct implementations. Modern 3D engines and robotics systems use concatenation millions of times per second for realtime computations of complex transform hierarchies.




More Quaternion Functions

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