From Rotation Matrices to Quaternions: Derivations, Efficient
Algorithms, and Applications in Computing and Physics
Chongjing Li
Guangdong Guangya High School, Nanyuan Street, Liwan District, China
Keywords: Rodrigues Rotation Formula, 3D Rotation, Quaternion Derivation, Efficient Algorithms, Applications in
Computing and Physics.
Abstract: This paper explores the mathematical foundations and applications of quaternions in representing three-
dimensional rotations. Quaternions, discovered by William Rowan Hamilton in 1843, provide a powerful and
efficient alternative to traditional rotation matrices. The paper begins by deriving the rotational quaternion
from rotation matrices, highlighting the mathematical relationship between the two. It then compares the
advantages and disadvantages of using quaternions versus rotation matrices for 3D rotations, emphasizing the
efficiency and numerical stability of quaternions. The paper also introduces new algorithms for reducing the
computational complexity of quaternion operations while maintaining accuracy. Finally, it discusses the
extensive applications of quaternions in computer animation and modern physics, demonstrating their
versatility and importance in enhancing computational efficiency and realism. By addressing both theoretical
and practical aspects, this paper aims to provide a comprehensive overview of quaternions and their
significance in various fields.
1 INTRODUCTION
William Rowan Hamilton, an Irish mathematician, is
renowned for his groundbreaking discovery of
quaternions. The story of this discovery is both
fascinating and emblematic of a moment of genius. In
the early 1840s, Hamilton had been deeply engaged
in the study of complex numbers and their geometric
interpretation in two dimensions. He sought to extend
this concept to three dimensions by finding a way to
multiply "triplets" of numbers, but he faced
significant challenges.
On October 16, 1843, Hamilton and his wife were
taking a walk along the Royal Canal in Dublin,
heading to a meeting of the Royal Irish Academy. As
they crossed Brougham Bridge (now known as
Broom Bridge), Hamilton experienced a sudden flash
of insight. He realized that the solution lay not in three
dimensions, but in four. He conceived the
fundamental formula for quaternion multiplication:
𝑖

=
𝑗

=𝑘

= 𝑖𝑗𝑘 = −1
(1)
This formula defined a new algebraic system where
multiplication was non-commutative, meaning that
the order of multiplication mattered. Excited by his
discovery, Hamilton immediately carved this
equation into the stone of the bridge. This act of
spontaneous graffiti has since become a celebrated
moment in the history of mathematics.
Hamilton's quaternions revolutionized the field by
providing a powerful tool for describing three-
dimensional rotations and spatial transformations.
Although quaternions were initially met with
skepticism due to their departure from traditional
algebraic rules, they eventually found wide
applications in various fields, including computer
graphics, robotics, and quantum physics. Today, the
discovery is commemorated by an annual "Hamilton
Walk" that retraces his steps along the Royal Canal.
This article is mainly about the summary of previous
work done on the concept of quaternion and rotation,
which includes the derivation of quaternion by using
rotational matrix, the comparison between using
quaternion and rotational matrix in representing 3D
rotation, new algorithm for reducing the complexity
of quaternion calculation, and the application of
quaternion in computer animation and modern
physics. (Kuipers, 2002). Hanson (2006) delves into
quaternions, offering a comprehensive visualization
analysis that aids in understanding their
representation of rotations in three-dimensional space.
342
Li, C.
From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics.
DOI: 10.5220/0013825500004708
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy (IAMPA 2025), pages 342-347
ISBN: 978-989-758-774-0
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.

Griffiths (2017) explores the application of
quaternions in quantum mechanics, particularly their
advantages in describing particle rotations. In the
realm of data processing, Fletcher and Joshi (2007)
discuss the application of quaternions in tensor data
analysis, showcasing their potential for handling
intricate data structures. Regarding robotic
applications, Sarabandi and Ha (2018) propose an
efficient quaternion-based computation method for
pose estimation. Lastly, Kim and Nam (2004)
introduce a quaternion-based interpolation technique
that significantly enhances the smoothness of
rotational transitions in computer animations.
The first section will be about introducing the
concept of quaternion and how it is applied to
represent 3D rotation. This section will first start from
using traditional method of rotational matrix to figure
out each component of a rotational quaternion. Then,
using the rotational quaternion and formula to show
how it represents 3D rotation. After the derivation,
the section will talk about the limitations of using
traditional methods in representing 3D rotations and
advantages of using quaternion.
The second section will be about introducing new
algorithm for reducing the complexity of quaternion
calculation. This is a relatively new development in
the field.
The final section will be about the application in
computer animation and modern Physics. It will talk
about how quaternion plays a role in representing
rotation when using cameras to shoot photos. Also, it
will cover how the quaternion was used in certain
fields in Physics such as quantum mechanics.
2 FIRST SECTION
This section will be about proving the rotational
quaternion by using rotational matrix.
Firstly, there are some prefix rules for quaternion
multiplication:
ii=
𝑗
𝑗=𝑘𝑘=−1
(2)
𝑖𝑗 = −
𝑗
𝑖=𝑘
(3)
𝑗
𝑘 = −𝑘𝑗 = 𝑖
(4)
𝑘𝑖 = −𝑖𝑘 = 𝑗 (5)
From the prefix rules above, we can see that the
quaternion multiplication satisfies the right-hand rule,
and it doesn’t satisfy the commutative law.
The basic form of a quaternion is shown below
𝑞=𝑎+𝑏𝑖+𝑐𝑗+𝑑𝑘
(6)
Where a is a real number and the rest of them are
imaginary number.
From the basic form above we can write down our
rotational quaternion which is
𝑞=𝑐𝑜𝑠(


)+sin(


)𝑢
(7)
2.1 The Derivation of Rotational
Quaternion by Using Rotational
Matrix
Firstly, there is a 3×3orthogonal Rodrigues rotation
matrix, which is used to represent the rotation about a
unit vector 𝑢=(𝑢

,𝑢

,𝑢

) with the degree of 𝜃
Then, following steps can be made after using the
matrix
𝑞

=1/2

1+𝑅

+𝑅

+𝑅

(8)
𝑞

=1/4𝑞

(𝑅

−𝑅

)
(9)
𝑞

=1/4𝑞

(
𝑅

−𝑅

)
(10)
𝑞

=1/4𝑞

(𝑅

−𝑅

)
(11)
After getting these final results we can have
following steps and outcomes
𝑞

= cos
(
𝜃/2
)
(12)
𝑞

=𝑢

𝑠𝑖𝑛
(
𝜃/2
)
𝑖
(13)
𝑞

=𝑢

𝑠𝑖𝑛
(
𝜃/2
)
𝑗
(14)
𝑞

=𝑢

𝑠𝑖𝑛
(
𝜃/2
)
𝑘
(15)
𝑞=𝑐𝑜𝑠(𝜃/2)+𝑠𝑖𝑛(𝜃/2)𝑢
(16)
Then, the basic form of the rotational quaternion
formula which is
𝑢
’
=𝑝𝑣𝑝

(17)
For a pure quaternion v = (0,𝑣

,𝑣

,𝑣

)
From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics
343

𝑣
→
=𝑣
→
↑↑
+𝑣
→

(18)
v
→
↑↑
=(
v
→
⋅u
→
)⋅u
(19)
𝑣
→

=𝑣
→
−𝑣
→
↑↑
(20)
𝑞𝑣
→

𝑞
∗
=𝑐𝑜𝑠𝜃𝑣
→

+𝑠𝑖𝑛𝜃(𝑢
→
×𝑣
→

)
(21)
𝑣
→
,
=𝑣
→
↑↑
+𝑐𝑜𝑠𝜃𝑣
→

+𝑠𝑖𝑛𝜃(𝑢
→
×𝑣
→

)
(22)
This result agrees with the formula we need to
prove.
2.2 The Comparison of Quaternion and
Rotational Matrix in Representing
3D Rotation
Rotation matrices are a widely used mathematical
tool for representing three-dimensional rotations due
to their intuitive nature and compatibility with
computer graphics and other applications. A rotation
matrix is a 3×3 orthogonal matrix with a determinant
of 1, which directly represents the rotation operation
and allows for straightforward visualization and
computation. This makes it particularly suitable for
applications where transformations need to be
combined with other operations such as translation
and scaling, as it can be easily integrated into a 4×4
homogeneous transformation matrix. Additionally,
the inverse of a rotation matrix is simply its transpose,
simplifying computations and ensuring numerical
stability.
However, rotation matrices have several
drawbacks. They require storing nine elements, even
though only three parameters are needed to describe
a rotation, leading to inefficient storage and
computation, especially in scenarios involving
frequent rotation combinations. Matrix multiplication
for rotation matrices is computationally intensive,
requiring 27 floating-point operations for a 3×3
matrix multiplication. Furthermore, when used in
conjunction with Euler angles, rotation matrices can
encounter the gimbal lock problem, which results in
the loss of a degree of freedom. Lastly, interpolating
between rotation matrices, such as for smooth
transitions in animations, is challenging and often
requires conversion to alternative representations like
quaternions.
Despite these limitations, rotation matrices remain
a valuable tool in applications where intuitive
visualization and compatibility with existing systems
are prioritized.
Quaternions offer several advantages for
representing three-dimensional rotations. They
require storing only four elements, making them more
storage-efficient than rotation matrices, which store
nine elements. This efficiency extends to
computational operations, as quaternion
multiplication involves only 16 floating-point
operations, compared to the 27 operations required
for 3×3 matrix multiplication. Quaternions also avoid
the gimbal lock problem, ensuring full representation
of rotations in three-dimensional space. Additionally,
quaternions provide a numerically stable method for
representing rotations, even after multiple
combinations of rotations. Interpolation between
quaternions, such as spherical linear interpolation
(Slerp), is straightforward and efficient, making them
ideal for applications requiring smooth transitions,
such as animations and simulations. Furthermore,
quaternions inherently maintain unit length when
normalized, which helps in preserving rotational
integrity during computations. (Shoemake, 1985)
However, quaternions also have notable
disadvantages. Their mathematical concept is more
abstract compared to rotation matrices, making them
less intuitive for beginners. The geometric
interpretation of quaternions is not as straightforward,
which can complicate debugging and visualization.
Quaternion operations with vectors require
converting vectors into quaternion form, performing
quaternion multiplication, and then converting back
to vector form, adding complexity to the process.
Additionally, converting between quaternions and
Euler angles involves intricate calculations, which
can be cumbersome in applications where Euler
angles are preferred. Lastly, while quaternions are
numerically stable, they require careful handling to
avoid issues like double rotations or sign ambiguities.
Despite these challenges, quaternions remain a
powerful tool in applications where computational
efficiency, interpolation capabilities, and avoidance
of gimbal lock are critical. They are particularly well-
suited for fields such as computer graphics, robotics,
and aerospace engineering, where frequent rotation
combinations and smooth transitions are essential.
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2.3 New Algorithms for Computing
Quaternion and Rotation
There are a few new algorithms for reducing the
complexity of quaternion multiplication and maintain
the accuracy of its outcome. Also, there are other
algorithms for simplifying storage structure and
enhancing interpolation efficiency. They serve for the
same goal but can be used for different purposes.
2.3.1 Algorithms for Reducing the
Complexity
Optimizing quaternion multiplication is essential for
enhancing the efficiency of 3D rotation calculations.
One effective approach involves “exploiting
symmetry and mathematical identities”. By
precomputing intermediate terms such as
𝑞1

𝑞2

,𝑞1

𝑞2

, and others, redundant calculations
can be minimized, thereby reducing the total number
of floating-point operations required for quaternion
multiplication. This method leverages the inherent
symmetry in the multiplication process to streamline
computations.
Another optimization strategy involves the use of
approximation algorithms. In scenarios where
minimal accuracy loss is acceptable, such as in real-
time graphics or simulations, approximation methods
like Taylor series expansion or polynomial
approximation can replace exact calculations. These
algorithms estimate complex functions involved in
quaternion operations, significantly reducing
computational overhead while maintaining
acceptable precision.
Incremental update strategies are particularly
useful in applications requiring frequent quaternion
updates, such as animations or physical simulations.
By adjusting quaternion values incrementally and
controlling the size of these increments, the need for
full quaternion multiplication and normalization can
be minimized, thereby reducing computational load.
Parallel computing represents a powerful
optimization technique for quaternion multiplication.
By distributing the computation of quaternion
components across multiple processing units, such as
GPUs or multi-core CPUs, the overall computation
time can be significantly reduced. This approach
leverages the parallel processing capabilities of
modern hardware to accelerate quaternion operations.
Lastly, storage optimization can enhance
computational efficiency by reducing memory access
overhead. Since quaternions are unit-length, storing
only three components and deriving the fourth when
needed can save memory and improve access
efficiency. This method is particularly beneficial in
resource-constrained environments.
These optimization methods collectively enhance
the efficiency of quaternion-based 3D rotations,
making them more suitable for applications
demanding high computational performance. By
selecting the appropriate optimization strategy based
on specific requirements, developers can achieve
significant improvements in both speed and resource
utilization.
2.3.2 Algorithms for Maintaining the
Accuracy of Quaternion Calculation
Enhancing the numerical stability of quaternion-
based 3D rotations is crucial for ensuring accurate
and reliable computations in various applications.
One effective approach is “incremental
normalization”, which adjusts the quaternion
incrementally to maintain unit length. This method
avoids the computational overhead of frequent full
normalization by making small adjustments during
each update step, ensuring the quaternion remains
close to unit length without significant computational
cost. This strategy is particularly beneficial in real-
time applications where efficiency is paramount.
Another strategy involves “ error correction
mechanisms” that monitor and correct deviations
from unit length. These mechanisms prevent error
accumulation by dynamically adjusting the
quaternion's components when deviations are
detected. By integrating these corrections into the
rotation update loop, numerical errors are minimized,
ensuring stable and accurate rotations over multiple
operations. This approach is especially valuable in
iterative algorithms where small errors can compound
and affect overall accuracy.
“Robust interpolation algorithms” also play a key
role in improving numerical stability. Traditional
interpolation methods like Slerp can encounter
instabilities, particularly when the angle between
quaternions is close to 0 or π. To address this,
advanced interpolation algorithms use polynomial
approximations or other techniques to handle these
edge cases more effectively. These methods ensure
smooth and stable transitions, even under challenging
conditions, making them suitable for applications
requiring high precision.
Finally, “hybrid representation methods” offer a
powerful solution by switching between quaternions
and other representations like rotation matrices or
Euler angles. This approach avoids singularities and
numerical issues by leveraging the strengths of
different representations based on the current rotation
From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics
345

state. Hybrid methods are particularly effective in
complex simulations and robotics applications where
singularities are more likely to occur.
By integrating these strategies, developers can
achieve stable and accurate quaternion-based
rotations, extending their applicability to demanding
scenarios from real-time animations to complex
simulations. These methods collectively address
common numerical challenges, ensuring reliable
performance in a wide range of applications.
3 THE APPLICATION OF
QUATERNION IN MODERN
PHYSICS AND COMPUTER
ANIMATION
Quaternions have become indispensable in modern
physics and computer graphics, offering efficient and
numerically stable representations of 3D rotations.
Their ability to avoid the gimbal lock problem
inherent in Euler angles makes them particularly
valuable in various scientific and creative
applications. In rigid body dynamics simulations,
quaternions provide a robust framework for
representing rotational motion. For example, in
simulations of celestial mechanics, quaternions are
used to model the complex rotational dynamics of
planets and moons, ensuring precise and stable
calculations that would be challenging to achieve
with traditional methods. This precision is crucial for
predicting celestial events and understanding the
behavior of astronomical bodies. (Chang, 2011)
In the realm of robotics, quaternions are employed
for real-time motion planning and control. Robots
often need to adjust their orientation quickly and
accurately, and quaternions facilitate these
adjustments by enabling efficient computation of
rotational paths. This is particularly important in
applications such as robotic arm control, where
precise orientation is essential for tasks like assembly
and manipulation.
In computer graphics, quaternions are pivotal for
creating realistic animations and interactions. In film
production, they are used to animate characters with
natural and fluid movements. For instance, when
animating a character performing a complex dance
routine, quaternions ensure that each movement is
smooth and realistic, from spins to leaps. Similarly,
in video game development, quaternions are used to
control camera movements, providing players with a
seamless and immersive visual experience. The
camera can smoothly transition between different
angles and perspectives, enhancing the overall
gameplay. (Shoemake, 1998)
Moreover, in virtual and augmented reality
applications, quaternions play a crucial role in
tracking and rendering. They are used in head-
mounted displays to track the user's head movements
accurately, ensuring that the virtual environment
responds instantly to the user's orientation. This real-
time tracking is essential for creating immersive
experiences where the virtual world aligns perfectly
with the user's movements. (Cutler & Eustice, 2010)
These examples illustrate the versatility and
importance of quaternions in modern computational
methods, highlighting their ability to enhance both
the efficiency and realism of simulations and
visualizations across diverse fields.
4 CONCLUSIONS
This paper delves into the concept of quaternions and
their pivotal role in representing three-dimensional
rotations, offering a comprehensive comparison with
traditional rotation matrices. We elucidate the
derivation of quaternions from rotation matrices and
highlight the advantages and disadvantages of both
methods. Quaternions, with their efficiency and
ability to avoid issues like gimbal lock, emerge as a
superior choice for many applications. We introduce
novel algorithms aimed at reducing the computational
complexity of quaternion operations while
maintaining numerical stability, which is crucial for
accurate and reliable rotations in various
computational tasks. Furthermore, we explore the
extensive applications of quaternions in modern
physics and computer graphics, demonstrating their
indispensable role in simulations, animations, and
virtual reality environments. These applications
underscore the versatility and importance of
quaternions in enhancing both the efficiency and
realism of computational methods across diverse
fields.
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