What is Quaternions: Definition and 82 Discussions

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.
Quaternions are generally represented in the form







{\displaystyle a+b\ \mathbf {i} +c\ \mathbf {j} +d\ \mathbf {k} }
where a, b, c, and d are real numbers; and i, j, and k are the basic quaternions.
Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by



{\displaystyle \mathbb {H} .}
It can also be given by the Clifford algebra classifications












{\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).}
In fact, it was the first noncommutative division algebra to be discovered.
According to the Frobenius theorem, the algebra


{\displaystyle \mathbb {H} }
is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. (The sedenions, the extension of the octonions, have zero divisors and so cannot be a normed division algebra.)The unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).

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  1. cianfa72

    I Is the Belt Trick Possible with Continuous Deformation in 3D Rotation Space?

    Hi, in the following video at 15:15 the twist of ##4\pi## along the ##x## red axis is "untwisted" through a continuous deformation of the path on the sphere 3D rotations space. My concern is the following: keeping fixed the orientation in space of the start and the end of the belt, it seems...
  2. Codezero

    I Modeling Asteroid Rotation Using Quaternions: Seeking Guidance on Init

    Hello everyone, I am an International Baccalaureate (IB) student working on my extended essay, which is a mandated 4,000-word research paper. My chosen topic is Quaternions, a mathematical concept I find highly intriguing. The primary aim of my paper is to model the rotation of an asteroid...
  3. jonagad

    I Error in Euler angles and quaternions

    Hi, I got a set of Euler angles and a set of quaternions, and I wanted to compare each set against its corresponding set obtained from STK, and I was wondering what would be a good indicator to measure the error between the Euler angles I got and those from stk , and the same for quaternions...
  4. N

    Rotating object using product of two quaternions

    Hello guys, I'm a newbie. So I have developped an application that rotates a cube using quaternion. The initial values of the quaternion are ( w=1.0, x=0.0, y=0.0, z=0.0). Now I want to apply two consecutive rotation using two different quaternion values: The first rotation corresponds to...
  5. M

    Algebra What's a good first textbook on Quaternions?

    Thanks to another thread I created, I already know what pre-requisite math subjects to study, and in what order to study them, before I'm ready to start studying Quaternions. I'm just very curious about what specific textbook, would you folks on this forum recommend that I get to study...
  6. J

    MHB Practical issue regarding Quaternions

    Hi there! I´ve got a practical problem with quaternions which I was not able to solve by my own so far. A machine detects the position and orientation of an object which I get as unity quaternion. I visualize that using matlab, which still works more or less, but I´d like to 'force' the...
  7. J

    Quaternions and Direction Cosine Matrix changing in time

    I've already posted this question on the mathematics website of stack exchange, but I've received more help here in the past so will share it here as well. I am developing a tool for missile trajectory (currently without guidance). One issue is that the aerodynamic equations on the missile are...
  8. Roni BM

    I How to derive Non-normalized quaternion with respect to time?

    I know that for normalized quaternion, $$\hat{q}$$, the derivative is given by $$\frac{d\hat{q}}{dt}=\frac{1}{2}\hat{q}\cdot \omega$$ where $$\cdot$$ denotes the quaternion multiplication. I want to calculate the time derivative of a non-normalized quaternion q. I tried to calculate the...
  9. jedishrfu

    B Cool Interactive Video on Quaternions

    Ben Eater's cool site has a couple of videos using Grant Sanderson's (3blue1brown youtuber) quaternion videos and making them interactive. WHile the discussion is going on you can manipulate the content. https://eater.net/quaternions/
  10. Ben2

    A Quaternions and Video Game Design

    On p. 566 (Section 563) of Hamilton's Lectures on Quaternions, I find "Operating with phi, and making reductions analogous to those of recent articles..." The 2nd derivation, beginning with "phi rho prime" so far totally eludes me. Let me apologize if this is the wrong forum. Quaternions have...
  11. K

    B Any geometrical meaning of multiplication of quaternions?

    Let's just talk about unit quaternions. I know that $$\left(\cos{\frac{\theta}{2}}+v\sin{\frac{\theta}{2}}\right)\cdot p \cdot \left(\cos{\frac{\theta}{2}}-v\sin{\frac{\theta}{2}}\right)$$ where ##p## and ##v## are purely imaginary quaternions, gives another purely imaginary quaternion which...
  12. Vectronix

    I Quaternions vs. Einstein's four-vectors

    Why did physicists abandon the use of quaternions? Can you tell me some of the differences between quaternions and Einstein's four-vectors used in special relativity?
  13. Math Amateur

    MHB Bresar, Lemma 1.3 - Finite Division Algebras .... real quaternions ....

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of the proof of Lemma 1.3 ... ... Lemma 1.3 reads as follows...
  14. Math Amateur

    I Bresar, Lemma 1.3 - Real Quaternions .... Division Algebras

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of the proof of Lemma 1.3 ... ... Lemma 1.3 reads as follows: In the above text by Matej Bresar we read...
  15. Fidelis

    A Finding the Angle Between Two Quaternions: A Simple Solution?

    Hello! I have two quaternions with norm equal to 1. Both are represented in the angle-phase form, i.e, I have q=exp(i*\phi)exp(k*\psi)exp(j*\theta) and p=exp(i*\phi')exp(k*\psi')exp(j*\theta'). Let \alpha be the angle between q and p. I need to write \alpha in function of \phi-\phi', \psi-\psi'...
  16. J

    Algebra Looking for a book on Quaternions

    Hello everyone, Lately, I have been reading and studying the Maxwell's https://es.wikipedia.org/w/index.php?title=A_Treatise_on_Electricity_and_Magnetism&action=edit&redlink=1 https://es.wikipedia.org/w/index.php?title=A_Treatise_on_Electricity_and_Magnetism&action=edit&redlink=1 Thanks for...
  17. EinsteinKreuz

    What is the mapping between the hyperriemann sphere and the complex plane?

    As many of you know, using the stereographic projection one can construct a homeomorphism between the the complex plane ℂ1 and the unit sphere S2∈ℝ3. But the stereographic projection can be extended to the n-sphere/n-dimensional Euclidean space ∀n≥1. Now what I am talking about is the the...
  18. EinsteinKreuz

    The biquaternions and friends

    So upon reading the wikipedia entry about the biquaternions I noticed that this algebra has several interesting subalgebras: 1. The split-complex numbers of the form {σ = x+y(hi)| ∀(x,y)∈ ℝ} which have the norm σ⋅σ* = (x2-y2). 2. The tessarines which can be written as {α + βj | ∀(α,β,)∈ℂ1 & j2...
  19. B

    Algebra Which books provide the best understanding of quaternions for scientists?

    I am very much interested in gaining an in-depth knowledge of quaternions, yet I cannot find any reviews of books on quaternions anywhere. Does anyone have any recommendations? Are Hamilton's and Tait's books my best bet?
  20. A

    Quaternion Derivative: Product Rule Explained

    How does the quaternion derivative work in the presence of a quaternion product. More specifically, does the standard product rule apply for quaternion derivatives? Say, I have a function f(q) = q* x a x q [where q -> quaternion, a -> const vector x-> quat prod] what is the result of the...
  21. Meditations

    Additional Phase factors in SU(2)

    I am curious as to the meaning of, and name given to the phase ##\xi(t)## which may be added as a prefix to the time evolution operator ##\hat{U}(t)##. This phase acts to shift the energy of the dynamical phase ##<{\psi(t)}|\hat{H}(t)|\psi(t)>##, since it appears in the Hamiltonian along the...
  22. Math Amateur

    MHB Properties of Unit Quaternions

    I am reading an article by Vernon Chi on quaternions and rotations in 3-space. The title of the article is as follows: I am concerned that I do not follow the proof of one of the properties of unit quaternions in Section 3.1.3 of the article. Section 3.1.3 reads as follows: In the above text...
  23. S

    Why use the notation (s, a, b, c) for quaternions instead of (s, v)?

    These are the notations of quaternions that i have seen: q = s + v q = (s, v) q = s + ai + bj + ck where s, a, b, & c are members of the reals but why not use the notation of: q = (s, a, b, c) isn't it the same as the 2nd notation except it is clearer? So why does it take a quaternion to be...
  24. twoski

    Can anyone help me fix quaternion rotation errors in Unity?

    I am trying to implement animation smoothing for a given set of keyframes. Some keyframes have "jitter" which i have to remove through my own algorithms. I am using the Unity game engine which has its own quaternion classes and such. I looked into it, and using an Exponential Moving Average...
  25. D

    Quaternions and Clifford Algebra problems

    Hello, I have some problems with understanding some concepts in Quaternions and Clifford Algebra. For example, where can I learn the basic construcion of Clifford Algebra? I'm listing the equalities I did not understand and I appreciate it if you can help me with understanding these : Homework...
  26. C

    Mapping Coordinate Systems Using Quaternions

    During the course of working with inertial measurement units (IMU) I have run into a problem. The issue is that an IMU reports accelerations relative to the IMU's orientation rather than it's initial orientation. The IMU's initial orientation is the identity quaternion (1,0,0,0). All changes...
  27. N

    Intuitive understanding of quaternions

    Complex numbers were fine. I understand that \imath=\sqrt{-1} . But what I am struggling with is the idea that j and k are somehow different in the context of quaternions. I don't know if j and k are also equal to \sqrt{-1} or something different since they do not commute. Also the idea that...
  28. S

    Exploring the Role of Quaternions in 3D Rotations

    So I have been heavily trying to understand rotations. Rotations as i understand is a planar phenomenon. You need at least two dimensions. That is why rotations cannot work in dim 1. With 2 dimensions, rotations happen in the only plane that exists: XY. However the axis of rotation cannot...
  29. M

    How is the algebra of quaternions isomorphic to the algebra of matrices?

    I just started learning about morphisms and I came across a problem that totally stumps me. Here goes: Show that the algebra of quaternions is isomorphic to the algebra of matrices of the form: \begin{pmatrix} \alpha & \beta \\ -\bar{\beta} & \bar{\alpha} \end{pmatrix} where α,β\inℂ...
  30. R

    Using quaternions to perform rotations

    Hi, I have been looking at quaternions to perform rotations, however I have come across two slightly different equations to do this: v' = q^{-1}vq v' = qvq^{-1} What is the difference between these two? Thanks, Ryan
  31. V

    Fastest interpolation method for attitude quaternions?

    I hesitated between posting this in the Mathematics forum or here, but since it's fairly applied, I chose this place. Sorry if it should've gone somewhere else. I posted another thread earlier (https://www.physicsforums.com/showthread.php?t=599737), about having trouble finding the quaternion...
  32. A

    Understanding Quaternions Multiplication: Exploring the Algebraic Properties

    Homework Statement Its not really homework problem, and you may find it silly because its only multiplication problem, but I cannot get the right answer by multiplying quaternions. Basically this is what i want to show: exp(iψ/2)exp(kθ/2)exp(iф/2) = cos(θ/2)exp(i[ψ+ф]/2) +...
  33. R

    IMU -> rotation angle -> quaternions

    I'm trying to determine the link between an IMU I have (tri-axis accelerometer, gyroscope, and magnetometer) and determining its rotation using quaternions. I've spent a while reading up on the IMU's properties, and quaternions, but I can't get my head around how the two meet. So...
  34. Math Amateur

    Centre of the real Hamilton Quaternions H

    Can anyone help me with the following exercise from Dummit and Foote? ============================================================ Describe the centre of the real Hamilton Quaternions H. Prove that {a + bi | a,b R} is a subring of H which is a field but is not contained in the centre...
  35. W

    Centre of the ring of quaternions

    Homework Statement What is the centre of the ring of the quaternions defined by: \mathbf{H}=\{ \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix} | a,b \in \mathbf{C} \}? Homework Equations The definition of the centre of a ring: The centre Z of a ring R is defined by Z(R)=\{A...
  36. T

    Composition of Quaternions as rotations

    Homework Statement Hi, I am having problems in showing that in practise the composition of two rotations represented by quaternions is still a rotation. The example I have constructed is: Rotate (1,1,0) by 45 degrees about the z axis. The quaternion to use is thus q = cos(22.5)+ksin(22.5)...
  37. T

    Rotations with Quaternions and their exponential.

    Homework Statement Hello, I'm trying to get my head around the various properties of quaternions that all seem very similar, but I can't quite understand the underlying differences between them. I would like to know the differences between unit quaternions, purely imaginary quaternions...
  38. J

    Eliminating the Euler Angle singularity without quaternions?

    Hi all, I've formulated using Lagrangian formalism the equations of motion for a spinning top. I know about the gimbal lock/singularity that occurs at theta=0 and I was wondering if there was any other way to do it without dwelving into quaternions. Yogi published a paper "A Motion of Top...
  39. Alain De Vos

    Einstein's Field equations & quaternions / octonions

    You can rewrite Maxwell's equations using d'Alembertian operator on quaternions. Can something similar be done for Einstein's Field equations and is there an advantage in doing so? Will this help in finding solutions to the equations or e.g. calculation of proper time,proper distance? Will...
  40. P

    Quaternions and rotation vector.

    Hi, I'm trying to calculate a normalized 3d vector representing the quaternion's orientation. Can anyone give me a hand? Thanks!
  41. lpetrich

    Generalizing the Quaternions

    Working with rotation matrices led me to consider quaternions, and that led me to consider the question of generalizing the quaternion group over different numbers of components of the quaternion vector. I've attempted to work out what possibilities there are, though someone else may have...
  42. S

    Where can I find resources to learn about quaternions?

    I am interested in learning more about quaternions because they can show more about a system than vector analysis. Does anyone know of a good website that teaches the theory and how to use quaternions? I have already tried the book Quaternions and Rotation Sequences. I might have to check it out...
  43. B

    Quaternions angle error - correct?

    Hi All, I think this is right, but not sure after doing some of the maths. If I have two rotated objects...lets say two sticks...and each has a rotation, in quaternions q0 and q1. Now the difference, can be calculated as qdiff = q0 * Conjugate( q1 ) Okay? Of course both my object...
  44. mnb96

    Quaternions and metric of the 3-sphere

    Hello, let's supppose I am given a unit-quaternion q expressed as an element of \mathcal{C}\ell_{0,2}(\mathbb{R}) as follows: \mathit{q} = a + b \mathbf{e_1} + c \mathbf{e_2} + d \mathbf{e_{12}} I now rearrange the terms in the following way: \mathit{q} = (a + d \mathbf{e_{12}}) +...
  45. T

    Ring Theory - Quaternions and sets of inverses

    Homework Statement Two questions really, the first is about the ring of quaternions H and the second about a set of maps. a) Find an element c in H such that the evaluation phi_c : C[x]-->H is not a ring homomorphism. In words that is: "the evaluation phi sub c from the ring of complex...
  46. mnb96

    Quaternions and associated manifolds

    Hello, it is known that pure-quaternions (scalar part equal to zero) identify the \mathcal{S}^2 sphere. Similarly unit-quaternions identify points on the \mathcal{S}^3 sphere. Now let's consider quaternions as elements of the Clifford algebra \mathcal{C}\ell_{0,2} and let's consider a...
  47. mnb96

    Exploring Unit Quaternions and Their Relationship to Spheres

    Hello, I read somewhere that the set of unit quaternions identifies the \mathcal{S}^3 sphere. This makes sense; however, what happens if we consider instead a quaternion as an element of the even-grade subalgebra \mathcal{C}\ell^+_{3,0} ? Now a unit quaternion is represented as a...
  48. A

    Complex numbers and hamilton quaternions generate [tex]M_{2}(C)[/tex]

    How can M_{2}(\mathbb{C}) be written as a combination of elements of \mathbb{C} and elements of \mathbb{H}?
  49. mnb96

    Quaternions in Clifford algebras

    Hello, it is known quaternions are isomorphic to \mathcal{C}\ell^{+}_{3,0}, which is the even subalgebra of \mathcal{C}\ell_{3,0} Is it possible to find an isomorphism between \mathcal{C}\ell_{2,0} and \mathbb{H} \cong \mathcal{C}\ell^{+}_{3,0} ? *** my attempt was: *** Let's consider...