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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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Robotics, Sensors and Physics: quaternions, velocity, acceleration [on hold]

Problem: Feature engineering Context: A robot with many sensors is moving around. I have this features: orientation (quaternions X,Y,Z,W) angular velocity (X,Y,X) linear acceleration (X,Y,Z) <...
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67 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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24 views

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
36 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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24 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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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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45 views

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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14 views

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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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
40 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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26 views

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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20 views

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
27 views

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
44 views

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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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
68 views

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
35 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
142 views

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
40 views

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
38 views

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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69 views

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
58 views

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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76 views

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
32 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
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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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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 ...
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1answer
30 views

How do I express a unit quaternion in exponential form? [closed]

Let $t,u,v$ lie in the interval $(-\pi, \pi]$ If we assume that $$ \cos(t)\cos(u)\cos(v) + \sin(t)\cos(u)\cos(v)i + \sin(u)\cos(v)j + \sin(v)k = e^z$$ such that $z$ is also a quaternion, what does z ...
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What's the power of quaternion $[e^{a} * (cos(r) + \frac{sin(r)}{r}(bi+cj+dk)) ]^ 2$?

Euler's identity extended into quaternions is: $q = a + bi + cj + dk$ with a,b,c,d real numbers for the below: $\sqrt{b^2+ c^2 + d^2} = r > 0$, and $\frac{bi+cj+dk}{r}$ = $\sqrt{-1}$, ...
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1answer
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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 $\Bbb R$, say $H(R)$. I want to show that $q \in H(R)$ is a unit iff $N(q)$ is a unit in $R$. If $q=a+bi+cj+dk$, ...
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2answers
72 views

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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1answer
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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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1answer
47 views

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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1answer
40 views

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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108 views

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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2answers
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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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20 views

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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1answer
36 views

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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1answer
92 views

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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1answer
45 views

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 ...