What is Quaternion: Definition and 46 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).

View More On Wikipedia.org
  1. 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...
  2. N

    Understanding Quaternion to DCM Conversion in Openshoe Matlab Library

    Summary:: Conversion from quaternion to DCM Hi All , our teacher asked us to try to understand the openshoe Matlab library , i stagnate on function that convert the quaternion to DCM i have found many example on website but the description of matrix are diffrente that the one is used on...
  3. DartomicTech

    What math course is Quaternion math taught in?

    Summary:: I want to learn Quaternion math. Also; I do not know what prefix my question falls under. I just learn maths from books. I don't know which parts of school they are taught in. I was hoping that I would get to learn about that subject in my Linear Algebra textbook, but I looked...
  4. 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...
  5. 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...
  6. Ben2

    A Can anyone alive derive these quaternion equations?

    If anyone has read Hamilton's Lectures on Quaternions, lines 13 and 15 of Section 563 (p. 566) have successfully resisted my efforts for months. For those interested, I can provide the Latex version. Let me also apologize if this is the wrong forum for the question. Thanks.
  7. S

    I Quaternion conversion extrinsic to intrinsic

    Hi All, I Have a system which is supplying me with quaternions, working in opengl I am setting the orientation of a model to the quaternion I am given, and it is seen that all changes in pitch are shown as changes in rotation around the opengl x-axis (1 is left), all changes in roll are shown...
  8. 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...
  9. J

    Can quaternion group be represented by 3x3 matricies?

    Hi, The Quaternion group, ##Q=\{1,-1,i,-i,j,-j,k,-k\}##, can be realized by ##2x2## matricies: ## \begin{align*} 1=\begin{bmatrix} 1,0 \\ 0,1\end{bmatrix} &\hspace{10pt} i=\begin{bmatrix} \omega,0 \\ 0,-\omega\end{bmatrix} & \hspace{10pt}j=\begin{bmatrix} 0,1 \\ -1,0\end{bmatrix} &...
  10. Topolfractal

    Quaternions: Extension of Real Numbers?

    Quaternions are generalizations of 3- vectors, in the same as complex numbers are generalizations of 2- vectors. Should quaternions be considered an extension of the real numbers as the complex numbers were?
  11. Topolfractal

    Quaternion derivatives

    I am trying to work out some basic aspects in the theory of quaternions for some work in physics I'm doing. I have went through complex analysis and saw that the only way division ( and hence the derivative) could be defined was through a numerical definition of (i). My question is does there...
  12. S

    Question in quaternion multiplication

    Hi guys, I am taking this class in lie groups but the professor never introduced the concept of quaternion and he asked about it. I saw from google the properties of multiplications of j and I made the multiplication according to (B + jC)(u + jv) = Bu + Bjv + jCu + j^2Cv = Bu - Cv + j(Cu + Bv)...
  13. T

    Quaternion in Image Processing - Learn How to Represent 3D Space

    Hi everyone I have question about Arithmetic quaternion. this concept is used in image processing for representing R G B color channel as a hyper complex number or single unit.how can it is possible? can I represent every things in 3D space as a hyper complex number by quaternion? Please...
  14. L

    Quaternion Group Multiplication Table

    Homework Statement Obtain multiplication table for quaternion group. Homework Equations ##i^2=j^2=k^2=ijk=-1## The Attempt at a Solution I have problem with elements for example ##ji## in the table. For example when I have ##ij## I say ##ijk=-1## and ##k^2=-1## so ##ij=k##. But...
  15. R

    Quaternion local rotation

    Hey, Once again, I got a question about quaternions. Say I have an initial rotation Q1. I now want to rotate Q1 on the X and then on the Y axis. BUT: The Y rotation should apply to the local Y axis which was given in Q1. The problem is: If i rotate Q1 by the X-rotation Q2, then the Y...
  16. R

    Homomorphisms of Quaternion Group

    Homework Statement Let Q = {±1, ±i, ±j, ±k} be the quaternion group. Find all homomorphisms from Z2 to Q and from Z4 to Q. Are there any nontrivial homomorphisms from Z3 to Q? Then, find all subgroups of Q. Homework Equations The Attempt at a Solution I don't even know...
  17. J

    Need Help with Calculating Quaternion Derivatives for Rigid Body Dynamics?

    Quaternions are new to me, so I constructed a simple model to help grasp the concept. I have a very simple dynamic model that used Euler's equations for the rigid body dynamics. The model only considers attitude; translational motion is ignored. I am making use of quaternions to describe the...
  18. Math Amateur

    MHB Generalised Quaternion Algebra over K - Dauns Section 1-5 no 19

    In Dauns book "Modules and Rings", Exercise 19 in Section 1-5 reads as follows: (see attachment) Let K be any ring with 1∈K whose center is a field and 0 \ne x, 0 \ne y \in center K are any elements. Let I, J, and IJ be symbols not in K. Form the set K[I, J] = K + KI + KJ + KIJ of all K...
  19. Math Amateur

    MHB Generalised Quaternion Algebra over K - Dauns Section 1-5 no 18

    In Dauns book "Modules and Rings", Exercise 18 in Section 1-5 reads as follows: (see attachment) Let K be any ring with 1∈K whose center is a field and 0 \ne x, 0 \ne y \in center K are any elements. Let I, J, and IJ be symbols not in K. Form the set K[I, J] = K + KI + KJ + KIJ of all K...
  20. Math Amateur

    MHB Generalised Quaternion Algebra over K - Dauns Section 1-5 no 17

    In Dauns book "Modules and Rings", Exercise 17 in Section 1-5 reads as follows: (see attachment) Let K be any ring with 1 \in K whose center is a field and 0 \ne x, 0 \ne y \in center K any elements. Let I, J, and IJ be symbols not in K. Form the set K[I, J] = K + KI + KJ + KIJ of all K...
  21. Z

    Reverse Quaternion Kienmatics equations

    Hi all!, first post! I'm programming an attitude estimation and control algorithm for a satellite. So I need a reference "trayectory" for my control system to try to follow. The generation of that attitude profile is tricky: point to the sun, if sat has access to a certain city, point to...
  22. S.Daedalus

    Quaternion Higgs and the LHC

    As everyone knows, since the fourth of July, the family of elementary particles has been re-united with its long-lost son, the Higgs boson. Of course, as every discovery, so this one, too, serves to open up further questions. The first one that presents itself is certainly: So, is this the...
  23. V

    Attitude quaternion derivatives from Euler angular velocities

    I'm struggling to understand what the derivative of an attitude quaternion really is and how to use it. I need it to solve a problem relating to a rotating frame of reference relative an inertial frame. The information I have is a vector of Euler angular velocities (i.e for roll \phi, pitch...
  24. D

    Quaternion Kaehlerian manifold, definition

    Hello, I am reading the paper of S. Ishihara, Quaternion Kaehlerian manifolds, I need it for understanding of totally complex submanifolds in quaternion Kaehlerian manifolds. I am afraid that I don't understand well the definition of quaternion Kaehler manifold, that is my question is the...
  25. A

    Quaternion Polynomial Equation

    I'm working on the following: "Prove that x^2 - 1=0" has infinitely many solutions in the division ring Q of quaternions." The Quaternions are presented in my book in the representation as two-by-two square matrices over ℂ. The book gives that for a quaternion (sorry for the terrible...
  26. B

    Problem with quaternion rotation maths

    Hi, I am trying to work out the maths to keep a camera connected to a balloon pointing in a fixed direction. The camera can roll, pitch and yaw relative to the balloon which itself can roll, pitch and yaw relative to an inertial frame. Sensors provide the camera roll, pitch and yaw relative to...
  27. S

    Solving Quaternion Confusion for 3D Orientation

    Hi Guys, I am getting a quaternion stream from motion sensor (accl, gyro, magneto). As I understand the quaternion represent the 3D orientation of objects. When I place the sensor at location 1 (I record quaternion, q1=[w x y z]), now if I place it at location 2 (say 3 feet to right, 2 feet...
  28. R

    A set the quaternion group could act on

    Homework Statement Find the smallest integer n such that the quaternion group G has a faithful operation on a set Sof order n. Homework Equations The Attempt at a Solution So the homomorphism between G and permutations of S is injective. which means the order of S_n is bigger...
  29. L

    Quaternion Rotations: Show R2∘R1 Is a Rotation

    Homework Statement 3. (a) Show that every quaternion z of length 1 can be written in the form z = cos(\alpha/2) + sin(\alpha/2)n, for some number α and some vector n, |n| = 1. (b) Consider two rotations of the 3d space: the rotation R_1 through \alpha_1 around the vector n_1 and the rotation...
  30. L

    Show that for every quaternion z we have

    Homework Statement Show that for every quaternion z we have: [tex] $ \overline{z} = \frac{1}{2}(-z-izi - jzj - kzk) $ [\tex]That is the question, I just don't know how to begin and the " izi - jzj - kzk " confuses me. I need help on how to start this. Thanks a lot :D
  31. S

    Quaternion derivative ambiguity

    In Quaternions and Rotation Sequences by Jack B. Kuipers (pg. 264-265) the quaternion derivative is defined as: \frac{dq}{dt}=q(t)\overline{\omega}(t) But in many published papers, I have seen the derivative defined instead as \frac{dq}{dt}=\frac{1}{2}q(t)\omega(t) Why is there a...
  32. S

    Quaternion conversion in satellite attitude using sun-earth sensors simulation

    I need to convert quaternion (q) to a form that is suitable to show changing attitude of a Satellite. like new x,y,z vectors of a satellite.I don't know the math of quaternions. I am getting updated state using rung -kutta 4 method where state vector x=[q(1:4) wx wy wz a(1:3) b(1:3)].I can...
  33. J

    Precise relation between quaternion and vectors

    I am reading "history of vector analysis": https://www.amazon.com/dp/0486679101/?tag=pfamazon01-20 We have two quaternions \alpha , \alpha' with "scalar" component equal to zero, these are their expressions: \alpha' = x' \vec{i} +y' \vec{j} + z' \vec{k} \alpha = x \vec{i} +y \vec{j} + z...
  34. S

    Understanding Quaternion Multiplication for 3D FPS Camera: A Simplified Approach

    I'm trying to look how the formula made on this http://www.gamedev.net/reference/programming/features/quatcam/page2.asp" . I want to know how this formula was derived. I tried to look on Quaternions and Rotation Sequences book but they use sin/cos which is so different on this. N.w = a.w*b.w...
  35. I

    Show that every quaternion z, where |z|= 1, can be expressed

    hi, and thanks for reading. hh, and this isn't homework, its just something I've been wondering about. i've been flicking through a linear algebra book, I'm trying to learn it by myself, and I've come across this question which has completely stumped me: show that every quaternion z, where...
  36. N

    What is the derivative of a quaternion with a changing unit vector?

    Hello everyone. Here's a unit rotation quaternion : q(t) = [cos\frac{\theta(t)}{2} , \hat{u}(t)\cdot sin\frac{\theta(t)}{2}] We know that if \hat{u}(t) is constant, then our quaternion's derivative should be : \dot{q}(t) = \frac{1}{2}\cdot q_\omega(t)\cdot q(t) But what if \hat{u}(t) wasn't...
  37. S

    Simple example of Quaternion intuition

    Why is it that I can't describe with words the orientation of a 3D object (i.e. I can't give a set of angles that uniquely describe it). On the other hand, I can mimic fairly precisely it's orientation with my hand to describe it. A one dimensional object however, is easy to describe with an...
  38. M

    Does Quaternion Calculus Extend Classical Complex Analysis Theorems?

    let be a quaternion a+ib+cj+dk and a,b,c,d are functions of (x,y,z,u) my questions are. - is there an anlogue of Cauchy integral theorem ?? , if an analytic function of a quaternion z , defined by f(z) , has a pole at the point 1+i+2j-3k How could you calculate its residue ?? - If a...
  39. G

    Proving Infinitely Many Solutions for u² = -1 in Quaternion Division Ring H

    Homework Statement Show that the quaternion division ring H has infinitely many u satisfying u^{2}=-1 Homework Equations Elements of H is of the form a.1 +bi+cj+dk where a, b, c, d in \textsl{R} ( reals) and i^{2}= j^{2}= k^{2}=ijk = -1. The Attempt at a Solution Let u = a.1...
  40. I

    When using Quaternion what is w ?

    When using Quaternion what is "w"? Hello, I think using Quaternion for my vector matrix rotation problem, but I can not find and explination of what the "W" parameter is/used for. Many thanks IMK
  41. robphy

    Double-Slit experiment in Quaternion Quantum Mechanics?

    I'm not sure if "Quantum Physics" (or "Beyond the Standard Model") is the appropriate place for this question. Please move if necessary. What does "Quaternion Quantum Mechanics" say about the famous double-slit experiment? Does it make the same quantitative prediction as standard [Complex]...
  42. L

    So, is there a quaternion complex calculus?

    "Quaternion complex calculus?".. Hello my question is if there exist an analogue of "complex calculus" for complex numbers but using "quaternions"..in fact if we define the Quaternion.. a+ib+jc+kd with the "commutation relations" [x_i ,x_j ]=\epsilon _{ijk} x_k then i would like to...
  43. H

    The product between quaternion

    For example: i^2=? j^2=? k^2=? ij=? jk=? ik=? ijk=? Is ij=ji? And how to prove them? And also,vector times vector, what is the product?
  44. Q

    Can Quaternion and Pauli Matrix algebra be linked in EM course?

    i am learning Quaternion now for my EM course. Can someone enlighten me on the correspondence between Quaternion and Pauli Matrix algebra?
  45. A

    The 3D equivalent to Quaternion?

    Hi all, Was wondering if the 3-dimensional equivalent to Quaternion has a name? And why does it seem like (at least for me) that only the groups, who’s number of values it holds is 2^n (where n is a integer value), are more intensively used compared to those who’s value count is not 2^n? I am...