Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [quaternions]

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

1
vote
1answer
16 views

Does every order in a quaternion algebra have a nice integral basis?

Let $A$ be a quaternion algebra over a number field $k$ and let $O \subset A$ be an order. Does there exist always a basis $\{1,i,j,k\}$ of $O$ over the ring of integers $\mathcal{O}_k$ of $k$ such ...
0
votes
0answers
10 views

Fundamental theorem of algebra and quaternions

I'm not sure if the fundamental theorem of algebra extends to every possible and imaginable numbers (real, complex, quaternions, etc.) but here's my question anyway. Let $f(x) = x^2-2ax+(a^2+b^2+c^2+...
0
votes
0answers
14 views

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 ...
0
votes
1answer
1k views

Transforming NED Acceleration Profile to Body Frame through Quarternions

I have an acceleration profile which is in the North-East-Down coordinate system. The moving object in question is 6 DOF, however, and frequently approaches 90 degrees in roll, pitch, and yaw, making ...
0
votes
0answers
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.
0
votes
1answer
28 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 ...
0
votes
0answers
8 views

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,...
-2
votes
1answer
46 views

The dimension of a $\mathbb{R}$-algebra

Let $D$ be an irreducible $\mathbb{R}$-algebra in $M_{n}(\mathbb{C})$ that it implies $D$ has independent vectors as $\{A_{1},...,A_{r^{2}}\}$. Let $D$ is isomorphic to $\mathbb{H}$ (quaternions) ...
1
vote
0answers
45 views

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(\...
1
vote
2answers
53 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
0
votes
0answers
53 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 ...
1
vote
0answers
42 views

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−...
1
vote
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 ...
5
votes
4answers
6k views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
0
votes
0answers
12 views

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! ...
1
vote
0answers
24 views

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 ...
2
votes
1answer
50 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-...
1
vote
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). ...
1
vote
0answers
46 views

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 ...
0
votes
0answers
14 views

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 ...
0
votes
0answers
13 views

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\\ \...
1
vote
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 ...
1
vote
3answers
928 views

Representing rotations using quaternions

I'm learning Unity and came across a situation where rotations are represented as Quaternions. I've heard that they where used in computer graphics, but never had to use them until now. What I can't ...
2
votes
0answers
31 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: ...
2
votes
1answer
47 views

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 $2\times 2$ ...
1
vote
1answer
39 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 ...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
0answers
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) = ...
1
vote
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 ...
97
votes
7answers
7k views

Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
0
votes
1answer
24 views

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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 (...
20
votes
3answers
2k views

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 ...
1
vote
1answer
44 views

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 ...
0
votes
0answers
27 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.
1
vote
2answers
155 views

Can we describe quaternions using bra-ket in quantum mechanics?

For example, the rotation plus translation of a point using the language of quaternions is written as $Q(0,x,y,z)Q^* + T$ where $Q$ is the unit quaternion, $(x,y,z)$ is the point, and $T$ is some ...
3
votes
3answers
92 views

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=...
1
vote
0answers
29 views

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 ...
0
votes
0answers
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 $q_c$ ...
1
vote
2answers
47 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 ...
0
votes
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 ...
1
vote
1answer
256 views

Indefinite quaternion algebra over Q

Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Q: If we choose another isomorphism $...
1
vote
2answers
50 views

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 ...
0
votes
0answers
68 views

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-...
2
votes
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}^...
0
votes
1answer
758 views

Converting quaternions to spherical angles

Consider a situation where a beam is shot at a cube C from an arbitrary position P. The cube detects the angle of incidence relative to its $ x $ axis. The cube can be rotated and moved, and the ...
3
votes
1answer
70 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 $i^4 = 1$?
2
votes
1answer
46 views

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 $...