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Questions tagged [quaternions]

For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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Show that $q$ is a unit in $H(R)$ iff $N(q)$ is a unit in $R$

Problem: Consider the Quaternions over a general commutative ring $R$ instead of $\Bbb R$. I want to show that $q \in H(R)$ is a unit iff $N(q)$ is a unit in $R$. If $q=a+bi+cj+dk$, and $N(q) = a^2+b^...
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Quaternion product of three vectors: meaning of vector part?

$\newcommand{\i}{\mathbf{i}} \newcommand{\j}{\mathbf{j}} \newcommand{\k}{\mathbf{k}} \newcommand{\a}{\mathbf{a}} \newcommand{\b}{\mathbf{b}} \newcommand{\c}{\mathbf{c}} \newcommand{\R}{\mathbb{R}}$If ...
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Classification of subalgebras of composition algebras

Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ ...
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How/why does the ICP algorithm using quaternions work?

I've come across the Iterative Closest Point algorithm using quaternions (as described in "A Method for Registration of 3-D Shapes" by Besl and McKay) and I'm wondering, why it works. To me it seems ...
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Is it correct to say the field of complex numbers is contained in the field of quaternions?

I believe it is correct to refer to the complex numbers and their 'native algebra' as a field. See for example Linear Algebra and Matrix Theory, by Evar Nering. I assume the same may be said for the ...
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Quaternion angle calculation

I'm working on a programming project, in this project I'm receiving an angle as a quaternion value, I partially understand how they work but I don't find any math to get the values I need. What I ...
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Using Quaternion Extension of Eulers Formula what is $e^{qw} * e^{qw} = e^{qw2}$?

Knowing that for quaternions Euler's identity is: $q = a + bi + cj + dk$ with a,b,c,d real numbers $\sqrt{b^2+ c^2 + d^2} = r > 0$ $e^q = e^{a + r\sqrt{-1}} = e^ae^{r\sqrt{-1}} = e^a(...
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The quaternion division ring contains an infinite number of elements $ u $ satisfying $ u^2=-1 $

Show that the quaternion division ring contains an infinite number of elements $ u $ satisfying $ u^2=-1 $ I was trying to solve the above exercise on Page 133, Basic Algebra, Jacobson. Maybe it is ...
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Differentiate quaternion differential equation

I'm reading at the moment this paper, but I'm not clear with some steps the do: On page 5: When I differentiate $\dot{\mu}$ by $mu$ there should be also $ -M_q w_q$ in the A matrix at position 0,0? ...
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Sylvester equation over quaternion

How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + a\cdot(b+\overline{b}) + b \cdot \overline{b} $$ and calculate $Dx$. But it didn't help. ...
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Confused about rotation matrices

Applying a rotation matrix to a vector means shifting its coordinates to perform the rotation effect. Applying a rotation matrix to a model at the origin $(0,0,0)$ is not the same at performing a ...
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What are some examples of quaternions in rectangular and exponential form?

I've been interested in quaternions for their ability to represent positions and motions in spacetime in application to game development; I have searched for articles regarding representations of ...
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Is $\mathbb Z_3+\mathbb Z_3\mathbf i+\mathbb Z_3\mathbf j+\mathbb Z_3\mathbf k$ a local ring?

I consider the quaternion division ring on $\mathbb Q_3$: that is $$\mathbb H_{\mathbb Q_3}=\{a+b\mathbf i+c\mathbf j+d\mathbf k \mid a,b,c,d\in\mathbb Q_3\}$$ with $\mathbf i^2=\mathbf j^2=\mathbf k^...
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Smallest rotation angle between quaternions accounting for symmetry

I am trying to compute a similarity measure between the orientation of 3D objects accounting for symmetry invariance. I have a set of 3D objects, which are defined by a center of mass, and 3 unit ...
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1answer
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Deriving quaternions via the cross product?

I'm new to Lie theory. The cross product is the Lie bracket of the Lie algebra $\mathfrak{so}(3)$. But this Lie algebra also "belongs" to the unit quaternions $Sp(1)$. Moreover $\mathfrak{so}(3)$ ...
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How are these Quaternion Algebras isomorphic? $\Big(\dfrac{\mathit{a,b}}{\mathit{F}}\Big) \simeq \Big(\dfrac{\mathit{b,a}}{\mathit{F}}\Big)$

So let $\Big(\dfrac{\mathit{a,b}}{\mathit{F}}\Big)$ be a quaternion Algebra over a field $F$ with char $\neq 2$, and let $i,j$ be the standard generators for the quat. Algebra, meaning $i^2=a$ and $j^...
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Quaternion that transforms a point like a 2D angle

Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem. The algorithm I've implemented in 2D works as follows: Given the points $A (A_x, A_y)$ and $B (B_x, ...
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52 views

Rotation distance

right now i am working on a computer vision project and will recognize fiducial markers, i created a simulation testbed for benchmarking the precision of different methods. E.g. i do rotate the ...
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Further than the Sedenions?

So after the quaternions came to my knowledge, I wonder if you could go any further with the complexity. Turns out you can with the octonions(8D numbers) and sedenions(16D numbers). But are there 32D ...
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Determining Maximal Orders of a Quaternion Algebra

Let $F=\left(\frac{a,b}{K}\right)$ be a quaternion algebra for some number field $K$. What method do we use to determine a maximal order of $F$ (or the class of maximal orders of $F$)?
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Elementary proof for even numbers of ramified places for a quaternion algebra over $\Bbb Q$

I want to explain to someone that for any quaternion algebra $H$ over $\Bbb Q$, the number of places of $\Bbb Q$ that $H$ is ramified is even. But he only knows basic of algebraic number theory, so I ...
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isomorphism between subset of SU(2) and SO(3)

I know that there is a surjective map $\Phi : SU(2)\to SO(3) $. My question is if there is a subgroup $A \subset SU(2)$ such that $\Phi_{|A}:A\to SO(3)$ can be a (group) isomorphism. What would $...
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A bit confused on Quaternion

Sorry for noobish question. I am trying to make use of quaternion in my work. I get input which has 2 parts: Corners of a square polygon, and a quaternion which encodes rotation of this polygon in ...
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Tensor Product over quaternions

I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as: $dq\,g \, \otimes \, d\bar{q}\, \bar{g}$ where $q$ is a general ...
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Is there an operation in complex numbers that can only be answered by quaternions?

The natural numbers cannot provide an answer to $1-2$. The integers cannot provide an answer to $\frac{1}{2}$. The rational numbers cannot provide an answer to $\sqrt{2}$. The real numbers cannot ...
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Exterior Derivative over Quaternions

I was wondering if it is possible to define the exterior derivative of a quaternionic valued function. I am doing the quaternionic analogue of a previously complex valued computation, namely something ...
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Using quaternion algebra, determine matrix of reflection given a plane

The question asked us to find the matrix of the reflection on the plane with equation $x+2y-2z=0$. From what I have learnt, the (unit) normal to the plane is $[\frac{1}{3},\frac{2}{3},-\frac{2}{3}]$. ...
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Factoring a quaternion expression with three variables

I have a quaternion-valued function of three real variables which I would like to rewrite as the product of two quaternions. The function is given by: $F(x,y,z) = r + 2 xy\ \textbf{i} + (x^2 + z^2)\ \...
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How to linearize Quaternions?

Based on an answer to one of my questions and the comments exchanged here earlier I noticed that I cannot uniformly sample Quaternion vectors for rotation even though if I know the bounds of each ...
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About Quaternion Isoclinic rotations

I have question about 4D-rotation using quaternion. I'm planning to use quaternion rotation in my game development this is as far as I understand. assume the target point of rotation P in 3Dspace ...
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Show that $G_E=\mathbb{S}^1\times \mathbb{S}^1$

I am stuck at the problem below. (I am using two identification $\mathbb{R}^4=\mathbb{H}$ and $\mathbb{S}=$ unit quaternion.) Show that $G_{E}=\mathbb{S}^1\times \mathbb{S}^1$ Firstly, giving ...
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does the inverse of a rotation quaternion result in the inverse of its matrix form?

according to wikipedia, one can get the inverse of a rotation matrix by simply using its transpose. Does this extend to rotation quaternions? Can I use the conjugate of a rotation quaternion, ...
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Shortest {angle,axis} between 2 frames which we know their quaternion with respect to a world frame

I want to figure out what is the shortest angle between 2 frames (a frame is an orthonormal set of 3 vectors, all starting from a same point), assuming they have rotated around the same origin. I ...
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The induced map $f' : S^3\times S^3\rightarrow SO(4)$ of $f : S^3\times S^3 \rightarrow GL_4(\mathbb{R}) $ is a homomorphism.

I am wanting to Show that a map $f' : S^3\times S^3\rightarrow SO(4)$ induced from a map $f : S^3\times S^3 \rightarrow GL_4(\mathbb{R}) $ is a homomorphism where $S^3$ is a Lie group of unit ...
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Left (Right) Linear Independence in Quaternion Matrix

I have a problem with understanding the concept of Left (Right) Linear Independence in Quaternion Matrix. I was studying the article 'Quaternions and Matrices of Quaternions*' by Fuzhen Zhang and I ...
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What is the equivalent of a (non-identity) matrix in quaternions

I'm using quaternions to solve Euler's equations of motion. I have the substitution that $\dot{q} = \frac{1}{2}q\bigodot\omega$, where $\omega$ is the angular velocity of the rotating frame. Hence $\...
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How to populate C values for quaternion

I am trying to work out the appropriate tilt calculations for a 3 axis accelerometer, I have tried many formulas using atan or atan2. The obvious issue with atan is I only get values -90/+90. With ...
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What is the valid range for elements of a Quaternion vector to be used for rotation?

I have seen different people saying different things about this so I'm confused. Assuming that each element of the Quaternion vector represents a rotation along some axis, it does not make sense to be ...
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Quaternions statistics from IMU

I am struggling with a problem about quaternions and how to use them in my application. I spent few days looking at a solution for this, but cannot find any similar applications or example around. I ...
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1answer
364 views

How to convert Euler angles to Quaternions and get the same Euler angles back from Quaternions?

I am rotating n 3D shape using Euler angles in the order of XYZ meaning that the object is first rotated along the X axis, then Y...
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Is there a non-Abelian algebra such that sequences can be 'reconstructed' given constraints?

Disclaimer: I'm a software engineer trying to use mathematics to solve a problem. I only have a shallow understanding of (abstract) algebra. My terminology will probably be wrong. Please be gentle. I'...
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How can ijk be equal to -1 if each is a square root?

I'm learning about Quaternions for the first time and ran into something I can't quite understand. Supposedly i^2 = j^2 = k^2 = ijk = -1; but if i, j, and k are all square roots of -1, then shouldn't ...
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How to randomly draw Quaternions within a specific range of Euler angles for rotation?

I am pretty new to Quaternions so please bear with me. I want to draw random Quaternion samples so that their Euler angle equivalent would range within [-30, +30] ...
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Convert quaternion to a line equation

I have a following set of data: Plane equation Camera position give an $x$, $y$, $z$ Camera rotation given as a quaternion I need to find an intersection of a line going straight ahead from the ...
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Show that a non-abelian group of order 8 with a single element of order 2 is isomorphic to Q group

where Q is the quaternion group. Thanks ! Here are my thoughts : With the Lagrange's theorem, we have that the order of the subgroup divides the order of the group. But there is a single element ...
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Help understanding the complex matrix representation of quaternions

Using the basis $ B = \{1, j\}$, one can show that quaternions can be represented by 2x2 complex matrices as follows: \begin{pmatrix} z & w\\ -\bar{w} & \bar{z}\\ \end{pmatrix} I would like ...
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How to find change in angle when vertical Scale of depth axis is modified?

Let Q be the Quaternion that perform a rotation of angle degrees around some axis of rotation. Later I change vertical exaggeration of my Vertical axis (z) to some value VS, e.g. make the vertical (z) ...
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computing polynomials whose roots are the vertices of 4-polytopes of circumradius one interpreted as quaternions

Suppose we have the vertices of 4-polytopes which have been scaled so that their are on the unit sphere. Interpret them as quaternions. If these have been computed before, I'm looking for a ...
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Distance between quaternions, ignoring one axis of rotation

Disclaimer: first time using quaternions. I needed to determine the distance between a orientation given by quaternion $q_1$ and a target orientation $q_t$. Both quaternions are normalized. Therefore:...
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Let G = Q8 (quaternion group). Find the right cosets of H in G for H = < J > and H = < -I > [closed]

For H = < J > , H = {J, -J, I, -I}, and H*K = {L, -K, -L, K} For H = < -I >, would H be all of Q8?