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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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How to know what $q$ is when $p$ and $f(p)$ are known in $f(p) = qpq^{-1}$

I keep running into a problem when I try to figure out a generalized way of rotating about $(1, 0, 0)$ or $(0, 0, 1)$, then trying to do the equivalent of either pitch up or turn. I would have gone to ...
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Calculating point B given point A and Euler angles in 3D [on hold]

Given 3D cartesian coordinates for point A relative origo, OA = (XA,YA,ZA), Euler Angles from point A facing point B, roll-pitch-yaw order, and length of AB. How do I calculate coordinates OB = (XB,YB,...
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Lipschitz primes

A Lipschitz integer is a Quaternion with integer coefficients. The norm is defined as $N(a+ib+jc+kd)=a^2+b^2+c^2+d^2$ which is a multiplicative function $N:\mathbb H\to\mathbb R$, $N(\alpha\beta)=N(\...
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1answer
27 views

How to rotate unit vectors and unit quaternions by unit quaternions?

First of all, I might have to apologize for my half-knowledge and the noob-questions which are about to come, but here we go: So, from what I've read/watched/heard, it appears to be like follows: Any ...
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2answers
48 views

Is a quaternion a way to divide vectors? [on hold]

This is a naive question but I read popular science articles where it is stated that quaternions define vector division, without further explanations
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An attempt to improve Plücker coordinates

Consider a 2-dimensional subspace of 4-dimensional vector space (of quaternions). It is a line in projective 3-space $P^3$. Let $u,v$ be the following 3-vectors (or vector quaternions) $u=d+m$ $v=d−...
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1answer
65 views

The equation $\zeta(q)=0$ for $q$ a quaternion

I know there have been several attempts to define a theory of functions of a quaternionic variable. I would like to know if a coherent and satisfying definition of the "Riemann" zeta function exists ...
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31 views

Where does $\cos(pi/2)$ and $\sin(pi/2)$ come from in quaternion rotation? Can you provide a simple unit quaternion rotation example?

I have seen many methods of rotation such as $p' = qpq^{-1}$ and $q = \cos (pi/2),v \sin (pi/2)$ , but I became slightly confused by how a unit quaternion is performed around a vector line.
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How to move a point [x, y, z] in 3D space around a center [0,0,0], using quaternion readings from a sensor?

I've been browsing the web for far too long, and still can't find a solution to this issue for my student project. Mind you my math skills are not that great, so I'm desperate for any help I can get! ...
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Vector part of $q^* v r$, what does it mean?

It's not clear why the quaternions are closed under addition. All of the constructions I've seen make it clear why they're closed under multiplication, but not addition. Anyway, consider the ...
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1answer
47 views

Norm of Quaternion

Given a quaternion of the form, $$q= a + bi + cj + dk$$ Which is the norm of $q$? (1) $\sqrt{a^2+b^2+c^2+d^2}$ (2) $a^2+b^2+c^2+d^2$ This page from MathWorks says (1) but another page says (2). ...
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Finding quaternion, representing transformation from one vector to another [closed]

Intro. Previously, I've asked a question on converting rgb triple to quaternion. After that question I've managed to get unit quaternions. Since they were unit ...
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Euler angle in two given coordinates

I have two coordinates. One is standard and the other is arbitrary. A rotation is represented as ZYX in the standard coordinates. (yes, It is not exactly a Euler angle. I'm sorry about this) and ...
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1answer
49 views

Quaternion conjugation map is orthogonal linear transformation

The following concerns an exercise from an undergraduate level textbook on Lie groups, which provides an elementary proof of $SU(2)/\mathbb{Z}_2 \cong SO(3)$. Think of $Sp(1)$ as the group of unit-...
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Angular error (in Euler angles) through quaternions

I found this formula in some notes but I would like to have a reference (book, paper, etc.) to understand where it comes from. I know that it works only for small angles. $ \begin{bmatrix} \phi_e\\ \...
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1answer
70 views

Constructing quaternions - proof that square of each imaginary unit is -1

During construction of vector space of quaternions over real numbers I encountered a problem that I can't quite put my finger on. For the context: Hamilton a multiplication in plane that keeps ...
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Why convert Quaternion to Euler Angle

I've recently played with the IMU filter in MATLAB. When using their examples, they always plot the rotations by stating something alike this: ...
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1answer
38 views

Solve for two quaternions that transformed another quaternion

I have the following problem: $$ q_2 = q_aq_bq_1{q_b}^{-1} $$ All the $q$'s are quaternions and I want to solve for $q_a$ and $q_b$, given more than one $[q_1, q_2]$ pairs, the last term is the ...
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26 views

Matrices over quaternions make Hopf Algebra or not?

I am learning Hopf algebra now a days. I am still confused about it’s axioms. I don’t know how to define antipode structure. What are the basic rules to define it. ? Can any one help me to solve this ...
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1answer
25 views

Find the rotation from two sets of 3 vectors?

I have three linearly independent vectors ($\vec{a}$, $\vec{b}$, and $\vec{c}$) which have been rotated to three other linearly independent vectors $\vec{a}'$, $\vec{b}'$, and $\vec{c}'$. I would like ...
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33 views

Group Automorphisms of Norm 1 Groups in Quaternion Algebras

Let $A$ be a quaternion algebra over a field $K$ of characteristic zero. By the famous Skolem-Noether theorem, every $K$-algebra automorphism $\varphi \colon A \to A$ is of the form $$ \varphi(x) = ...
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Left (right) eigenvalues of quaternion Matrix

I have problem with calculating left eigenvalues for quaternion Matrices. Let's take a look at article 'Geršgorin type theorems for quaternionic matrices - Fuzhen Zhang': Click The right eigenvalues ...
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1answer
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How to create a Quaternion rotation from a forward- and up- vector? [closed]

I need the rotation Quaternion of an object, I have it's foward and up directions (as 3D vectors), so I thought it would be easy to create a Quaternion rotation from that, but I can't seem to get it ...
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Aligning to quaternions in different coordinate frames

I am trying to compute the rotation quaternion between two quaternions called, q_IMU and q_VZ. I am computing the rotating quaternion by the method seen in the first equation at a maxima for both ...
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17 views

How to fix banking-drift in 3d road/path-data

I'm working on something in Unity3D (the game engine) where I have to modify a path/road in 3d space. The path consists of a collection of points that each have a position (Vector3) and orientation (...
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If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go: 1-tuple: Real numbers. 2-tuple: Complex numbers. 4-tuple: Quaternions. 8-tuple: Octonions. The Wikipedia ...
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1answer
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How to rotate a path in 3D (computer sciences)

I'm working on something in Unity3D (the game engine) where I have to modify a path/road in 3d space. The path consists of a collection of positions and quaternion-orientations (the orientation ...
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1answer
42 views

Describing maximal orders in quaternion algebras.

In Dorman's paper, Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves, he considers the following situation: $K = \mathbb{Q}(\sqrt{d})$ where $d$ is a ...
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Quaternion algebra using analytic continuation

As for complex variables, do we use analytic continuation to find things like $sin(j)$, $i^k$, and so on? Is there another method or do these expressions even have values at all.
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If x,y are orthonormal vectors with quaternionic coefficients, can we say that $\bar{x},\bar{y}$ are orthonormal?

In Morton L. Curtis's book Matrix Groups, it is stated that the orthonormality of the rows of a quaternionic matrix implies the orthonormality of the columns (Chapter 2 Proposition 4). However, as ...
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Torque required to achieve a desired quaternion

I was hoping someone can either explain or direct me towards a source that can help me with the following problem (not for homework, more of a hobby). Given an object with a current quaternion $q_c$ ...
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3answers
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If we are handed the presentation $\langle i,j,k \mid i^2=j^2=k^2=ijk \rangle$ and nothing more, can we deduce that this is the quaternion group?

If we are handed the group presentation $\langle i,j,k \mid i^2=j^2=k^2=ijk \rangle$ and nothing more, can we deduce that this is the quaternion group? Nothing in this presentation tells us that $i^2=...
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1answer
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Existence of conjugation on a quaternion algebra given a separable subalgebra

In Vignéras' book Arithmétique des algèbres de quaternions, a quaternion algebra $A$ over a field $K$ is defined as a $4$-dim central algebra for which there exists a separable $2$-dim (necessarily ...
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2answers
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Interpolation between three-dimensional rotations

I have to define a continuous function $g: [0, 1] \rightarrow \mathrm{SO}(3)$ such that $g(0) = I$ and $g(1) = R$ (a given rotation). I know we can do this kind of interpolation using quaternions ...
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2answers
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Showing that there is not a global diffeomorphism between unit quaternions and $\mathrm{SO}(3)$

I am new to differential geometry. I have the following question: Let $\mathbf{Q}$ denote the set of unit quaternions. I already have proved using the implicit function theorem that $\mathbf{Q}$ is a ...
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Proving that unit quaternions are a 3 Manifold

I am very new to topology, and I am having trouble on how to prove if something is a manifold or not. The question states that: Let Q donate the set of unit quaternions (a) Show that Q is a 3-...
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1answer
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Quaternion Group: Determine that $i^4 = 1$.

Suppose we are given the following presentation of the quaternion group: $Q_8 = \langle i, j, k \ | \ i^2 = j^2 = k^2 = ijk\rangle$ Is it obvious that $i^4 = 1$?
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1answer
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Pre-image of element in quaternion algebra

Let $A$ be an indefinite quaternion algebra (e.g. $(2,5)_\mathbb{Q}$), let $M$ be a maximal order in $A$ and let $\Gamma$ be the Fuchsian group derived from $M$. We will denote the group of units of $...
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1answer
38 views

What was the Hamilton's initial problem that led to him inventing quaternion?

I read wiki topic about history of Quaternion, and it confuses me on why, according to Hamilton, there's a problem with multiplication of triple (i.e. 1+i+j), and somehow the quadruple (or Quaternion) ...
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1answer
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Derivation for angular acceleration from quaternion profile

Given a profile of unit quaternions $q(t)$ that represents the orientation of a body over time, I like to get the angular acceleration $\dot \omega (t)$. I tried to find a formula myself, but I get ...
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1answer
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derivative action of isometries on hyperbolic 3-space (upper half-space model)

Let $\mathcal{H}^2=\{z=x+iy\in\mathbb{C} : y>0\}$ be the upper half-plane and let $g(z)=\frac{az+b}{cz+d}$, $a,b,c,d\in\mathbb{R}$, $ad-bc=1$, be an orientation preserving isometry of $\mathcal{H}^...
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2answers
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Prove or disprove ring $\mathbb{C}\times \mathbb{C}$ and ring Quaternion $H$ are isomorphic

My attempt: let $f$ ($w$+$x$i+$y$j+$z$k) $=$ ($w$+$x$i,$y$+$z$i) then I tried proving it is a homomorphism. $f$ is a homomorphism under addition but fails to be a homomorphism under multiplication. ...
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Do the octonions contain infinitely many copies of the quaternions?

Note: by "infinitely many", I'm confident I always mean $\beth_1$ many herein. We can easily show the quaternions contain infinitely many copies of $\Bbb C$ because, given any unit vector $\in\Bbb R^...
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1answer
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Hasse's classification of Quaternion algebras (over number fields)

I am looking for the reference to Hasse's original publication classifying the Quaternion algebras over a number field $K$. The statement in the paper or book should read something like: ``Two ...
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Are $\mathbb{C}$ and $\mathbb{H}$ the only continuous number systems over $\mathbb{R}^{2}$ and $\mathbb{R}^{4}$ respectively?

When I say $\mathbb{C}$ is a number system over $\mathbb{R}^{2}$, I mean that the field of complex numbers is identified with the set of dilative rotations of the affine real plane ($\mathbb{R}^{2}$) ...
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Finding and Comparing 2 Sensors Rotations, with same reference frame but different initial Orientation

Let's say we want to Compare two different Arm (Humerus) Rotations (series of quaternions) and we do not care about space translation but only for rotation. To measure each rotation we use the same ...
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1answer
34 views

Quaternions - Prove that two quaternions map to the same R

I am given the following question: I need to prove that two quaternions map to the same Rotation Matrix in SO3 Space. It is demonstrated by this image: Let w be v here. I tried to work out the proof,...
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Finding quaternion $q_x$ in the following equation: $q_x q_1 q_x' = q_2$

Given that $q_1$ and $q_2$ are known quaternions.
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3answers
68 views

Finding the Quaternion that rotates a coordinate system to match another.

Let's say I want to figure out the orientation of my cell phone. Assume that the phone has two internal sensors that report orientation (a quaternion), but both are a bit unreliable, so I'd like to ...
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0answers
23 views

Unique multiplicative quadratic form on quaternion algebras

I want to prove, that the only multiplicative quadratic form $Q$ (so $Q(xy)=Q(x)Q(y) \forall x,y$) on a quaternion algebra $\Big(\dfrac{a,b}{F}\Big)$ is the norm $\mathrm{Nr}$, which is isometric to ...