Questions tagged [quaternions]

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

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Calculating coordinates using an offset from an entity in 3D space knowing its position and rotation.

I am working on a game mod for FiveM which allows players to see bullet impacts and their angle of impact, and am trying to implement 'attaching' this evidence to objects which are not static. I have ...
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2 votes
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120 cell generated from quaternions

The two quaternions $\omega={1\over 2}(-1,1,1,1)$ and $q={1\over 4}(0,2,\sqrt{5}+1,\sqrt{5}-1)$ generate a finite group under multiplication with 120 elements that form the vertices of a 600 cell, ...
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how to offset a rotation contained in a unit quaternion rotation from the origin of a rigid object.

I'm using Unity3D for a project. The way it handles sorting transformations is with a 3vector-unit quaternion-3vector "sandwich" (the 1st vector for position, the quaternion for rotation, ...
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1 vote
1 answer
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Real matrices commuting with quaternion matrices

This is an interesting question from a class I'm taking that I'm not really sure how to approach. It is known that we can define many embeddings of the quaternion algebra $\mathbb{H}$ in the real ...
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Quaternions for coordinate frame rotation

I am attempting to switch over from Euler matrices to quaternions. Normally I am presented with a desired final orientation vector in the form of a unit direction and the angles of rotation to create ...
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Find any or all Lipschitz quaternions corresponding to a given quaternion norm

In this answer, in response to the question "How to find $2+7𝑖$ from $53$?", user @Cocopuffs provided a reference to a Jacobsthal sum, which enables obtaining a Gaussian integer, based on a ...
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2 votes
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Normalizing a dual quaternion

I'm kind of confused about the normalization of a dual quaternion. From the texts (e.g. Kavan 2008), for a dual quaternion $\hat{q} = a + \varepsilon b $, the norm is defined as $$ ||\hat{q}|| = ||a|...
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Trying to find a nice basis to realize the quaternion mapping as a rotation matrix

Trying to find a nice basis to realize the quaternion mapping as a rotation matrix. The quaternions are a 4 dimensional division algebra over $\mathbb{R}$ where we label the standard basis vectors as $...
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Transforming the ternary cubic polynomial in to a quaternion cubic homogeneous polynomial Form

I have a ternary cubic polynomial of the form $$ P(X,Y,Z) = \sum_{i=0}^{m1}\sum_{j=0}^{m2}\sum_{k=0}^{m3} c_{i,j,k}X^iY^jZ^k$$ how would one write it in quaternion from i know the the answer should ...
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17 votes
5 answers
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Why can't rotations be represented by purely imaginary quaternions?

I imagine this question has a straightforward answer, but I haven't been able to think of it on my own. It's well-accepted that you can represent a rotation in three-space with a unit quaternion. In ...
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Rotate a Translation and a Quaternion around the z axis of an arbitrary pose by an angle theta

I need to implement a rotation in a program but it's 15 years I haven't used rigid body motion maths. I use poses that are described by a translation T and quaternion Q. Everything is expressed in the ...
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Left ideals of group algebra $K[\mathbb H]$

I've been studying Hopf-Galois theory, and while writing by myself certain example, I came up with this question: Let $K$ be a field, and $\mathbb H$ the quaternion group. Is $K[\mathbb H]$ the only ...
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Holomorphy for functions with quaternion variables.

Why we cannot define the notion of holomorphy for functions with quaternion variables ?
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2 answers
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What is the smallest 3D rotation to make the axes line up

A 3x3 rotation matrix is considered axis-aligned if it consists of only 1, -1, and 0. Given an arbitrary rotation matrix, what is the smallest rotation required to make it axis-aligned? For example, ...
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quaternion-base logarithm of a quaternion: formula

Suppose you have two quaternion: $$q_1=(a_1, \vec{v_1})=a_1+b_1i+c_1j+d_1k$$ $$q_2=(a_2, \vec{v_2})=a_2+b_2i+c_2j+d_2k$$ I want to compute the logarithm base $q_1$ of $q_2$: $\log_{q_1}q_2$. In ...
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Group algebra for quaternion group

I'm trying to understand Hopf Galois Theory, and I decided to try studying some example of a non commutative ring extension. The papers I've studied tell me that, for a strongly $G$-graded algebra $A$ ...
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2 votes
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Is there a residue sum formula in quaternionic analysis?

In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series. In general, sums can be evaluated by means of this ...
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2 answers
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How to find the derived subgroup of $Q_8$

Given the group $Q_8=\{ \pm1, \pm i, \pm j, \pm k \}$ is there an easy way to find its derived subgroup? I've tried by hand and this seems such a big task to do manually. After this, by thinking a ...
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Transformation of vector plus quaternion avoiding 4x4 matrix multiplication (using dual quaternion?)

We have a need to translate and rotate a 3d position with orientation (expressed as quaternion) [x, y, z, qx, qy, qz, qw] by a translation and rotation expressed as vector and quaternion [x', y', z', ...
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2 votes
0 answers
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What is so special about the quaternion group that it is the only non abelian group for which every subgroup of it is normal?

If we disregard direct products, we find that quaternion group is the only non-Abelian group which has every subgroup as normal. Is there some deep property responsible for this 'outstanding' property ...
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Single axis movement - fixing quaternion values from misaligned IMU axis

I performed some IMU data collection (obtaining a time-series of quaternion values) to measure simple movements occurring in a single axis. The movement is contained to a single axis, but the IMU axis ...
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1 vote
1 answer
62 views

Determine the Lie algebra of the unit-quaternions $S^3 \subset \mathbb{H}^*$

Determine the Lie algebra of the unit-quaternions $S^3 \subset \mathbb{H}^*$ and their left-invariant vector fields. Unfortunately I am struggling with quaternions. I computed the differential of left-...
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Do all isometries of the Euclidean space necessarily send shapes to congruent shapes?

Proposition: $\mathbb{S^3}$ can be decomposed into disjoint congruent circles Solution:The quaternions of unit length $q=(a,b,c,d)$ of unit length satisfy the equation $$ a^2 + b^2 + c^2 + d^2 =1$$ ...
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2 votes
1 answer
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How to compute the $j$-invariant corresponding to a given maximal order in $B_{p,\infty}$

Let $B_{p,\infty}$ is the rational quaternion algebra ramified at $p$ and $\infty$. By Deuring's correspondence, there is a one to one correspondence between maximal orders in $B_{p, \infty}$ up to ...
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  • 1,019
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1 answer
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Determine the conjugacy-classes in the unit-quaternions $S^3$

The question is already in the heading. Determine all conjugacy-classes of the unit-quaternions $S^3=\{h \in \mathbb{H} | \: \bar{h}\cdot h = 1\}$. My only idea to tackle this question would be: Since ...
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1 answer
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About the octonions [closed]

Given $a\in \mathbb R^3-\{0\}$, then the map $J_a: q\to q\frac{a}{|a|}$ is a complex structure of the quaternions $\mathbb H$. Now, what about the octonions $\mathbb O$? Do they similarly obtain a ...
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Justifying the equivalence of different definitions of quaternion group. [duplicate]

There is an exercise in a group-theory notes founded online: The Q in the exercise is the quaternion group defined as the group generated by 2 elements a, b with: $a^4 = 1$; $b^2 = a^2$; $b^{-1}ab = ...
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How to select the conditions to generate a group? [closed]

One way to define the quaternion group is to define it as the group generated by 2 elements $a, b$ with: $a^4 = 1$; $b^2 = a^2$; $b^{-1}ab = a^{-1}$. There are also other groups defined in a similar ...
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1 answer
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Show that $\{\pm1,\pm i, \pm j, \pm k, \frac{\pm1 \pm i \pm j \pm k}{2} \}$ is a subgroup of $SO(3)$.

$i,j,k$ are the things we use for quaternion expression. To show the set is a subgroup, I'm trying to use a subgroup test, which is $ab^{-1} \in \{\pm1,\pm i, \pm j, \pm k, \frac{\pm1 \pm i \pm j \pm ...
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1 vote
1 answer
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If $q \in S^3$, we can write $q = \cos \theta + I \sin \theta$ with $I \in S^2$.

Please read whacka's answer to this post since my question is related to the answer. I would like to solve the problem: If $q \in S^3$, write $q = \cos \theta + I \sin \theta$ with $I \in S^2$. Prove ...
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0 answers
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In quaternions, and other hypergeometrics, is sqrt in a^2=b considered multivalued? [duplicate]

In a number system with multiple nonreals, p2=1, q2=-1, and/or r2=0 should have multiple solutions, right? For example, in the quaternions, q could be i, j, or k. Is it multivalued then? And what ...
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Show that $H \mathrel{\unlhd} SU(2)$ contains matrices with all possible traces between $2$ and $2\cos \varphi_0 $.

Let $H \mathrel{\unlhd} SU(2)$ that contains an element $A$ such that $\text{tr}(A)=2\cos \varphi_0 \neq \pm2$. Use i),ii) to show that $H \mathrel{\unlhd} SU(2)$ contains matrices with all possible ...
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5 votes
1 answer
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Conjugation of quaternions induces an isomorphism $\rho : S^3/\{\pm 1\} \rightarrow SO(3)$. I have a proof, but need some explanations.

Let $\rho(q):\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a map defined by $q_1\mapsto qq_1q^{-1}$ for a unit quaternion $q$. $S^3$ is the group of unit quaternions. I know that $\rho(q)$ acts orthogonally ...
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  • 1,248
0 votes
1 answer
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Multiplication of a unit quaternion and a vector in $\mathbb{R}^3$

As we know, a $q$ in the group of unit quaternions can be represented as a $2 \times 2$ matrix. So, if $q = a+ib+jc+kd$, then its matrix representation would be $\begin{bmatrix} a+ib & c+id \\ -c +...
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1 vote
1 answer
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The conjugation action $\mathbb{H}^*\times \mathbb{H} \rightarrow \mathbb{H}$ restricted to unit-quaternions yields an orthogonal representation

Consider the action $\mathbb{H}^*\times\mathbb{H} \rightarrow \mathbb{H}, (h,h')\mapsto hh'h^{-1}$. Show that it preserves the orthogonal-decomposition $\mathbb{R}\bigoplus $Im$\mathbb{H}$, and ...
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2 answers
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How do I solve (a+bi+cj+dk)^(f+gi+hj+nk)?

I purposely skipped using e as a factor in the title because e is Eulers Number.I have found $$e^{(a+bi+cj+dk)}$$ and $$(a+bi+cj+dk)^n$$ but no way to combine them together. My current theory is ...
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0 answers
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About transforming angles from one coordinate system to another

Say I have a gimble's wanted direction (in elevation phi and azimuth theta) in coordinate system xyz. Now I'm rotating the gimbal it by r about the x axis, p about the y axis and q about the z axis to ...
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3 votes
2 answers
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Does there exist some isomorphism from $Q_8$ to $\Bbb Z_4 \times\Bbb Z_2$?

I am wondering if some direct decomposition exists for quaternion group. I think that I am mixing some things, but let me explain and ask for clarification, tips from your side to let me understand my ...
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1 vote
1 answer
97 views

The commutator subgroup of real quaternion division ring

Let $\mathbb{H}$ be the ring of real quaternions and let $\mathbb{H}^*$ be the set of units in $\mathbb{H}$. Denoted by $[\mathbb{H}^*,\mathbb{H}^*]$ the commutator subgroup of $\mathbb{H}^*$. I want ...
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1 vote
0 answers
72 views

Angle between two quaternions

I am struggling to get my head around what I believe should be a simple problem. I am getting a quaternion wxyz from an IMU (inertial measurement unit), I want to check the angle between the ...
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-2 votes
1 answer
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Atan2 to unit rotation quaternion [closed]

In 3D space, I have an angle theta representing the orientation of the object. This angle is the result of the atan2 function. Using theta, how can I derive the unit quaternion which represents the ...
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0 votes
0 answers
45 views

Runge–Kutta integration for quaternion kinematrics

How can I use the Runge-Kutta integration method for quaternions Kinematics (https://arxiv.org/pdf/1711.02508.pdf) Incremental rotation $\Delta \theta = \omega_n \Delta t$ $ Exp(\omega \Delta t) = q\{...
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0 votes
0 answers
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How to get a discrete direction from quaternion rotations

Say you have multiple quaternions (of the form q = w, x, y, z) that represent absolute rotations. How can you get a discrete direction from there such that if we see quaternion 0 for example we can ...
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3 votes
1 answer
79 views

Implementation of a constructive algorithm for Deuring's correspondence

Let $E_0$ be a supersingular elliptic curve. By Deuring's correspondence, $\text{End}(E_0)\simeq \mathcal{O}_0$ is a maximal order in the quaternion algebra $B_{p,\infty}$ over $\mathbb{Q}$ ramified ...
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1 vote
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Classification of unit group of a maximal order of a quaternion algebra

Let $B$ be a definite quaternion algebra and $O$ a maximal order of $B$. In proposition 5.3.1. of Vigneras' "The arithmetic of quaternion algebra", the author classifies $O^*$. But there's ...
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Get location of point in different reference frame using quaternions

This may be a trivial question but I am not very well versed with quaternions. I have a camera which has a location and orientation in world coordinates. The location is (x,y,z) and the orientation is ...
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How to map from one coordinate/rotation system to another?

I'm using an Augmented Reality SDK to define a polygon in 3d space. What the SDK gives me back is a list of poses (3 translation coordinates, and a rotation quaternion), say ...
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1 vote
1 answer
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Help with closing this novel proof of the equivalence of oriented Grassmannians $G^+(2, 4)$ and $S^2\times S^2$

I'm concluding my thesis on the Hopf fibration, in which I constructed it purely geometrically. In a nutshell, you can treat $S^3$ as Quaternions, and then those Quaternions as rotations $R_{(\cdot)}$....
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2 votes
1 answer
88 views

How to convert a quaternion in polar form, back to rectangular form?

After a lot of digging in google, I found this(eqn. 36, pg. 2849) paper, which gives a 'real' polar form of a quaternion. It states that, any quaternion q can be represented as $q=\vert{q}\vert{e^{i\...
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4 votes
2 answers
85 views

Use of the quaternion magnitude

I'm using quaternions to describe 3D rotations which necessitates that the quaternion is normalized. I'm getting more interested in quaternions and I tried to check what the quaternion magnitude ...
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