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

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Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
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1answer
15 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
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35 views

Quarternions from MPU and circumference of circles

First I should mention that my math skills are super basic. I do not understand formulas but I do understand pseudo code, C, C++, and other programming languages. I've been working on a electronics ...
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1answer
54 views

Complex number with 3 dimensions [duplicate]

I was looking back on complex analysis and asked myself: ''Why is there no complex number in 3 dimensions ?''. To place this question let me define with what I mean with 3 dimensions in the following. ...
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1answer
17 views

Rotation orientations in n-dimensions

I'm doing a change of variables that involves doing simple rotations on the standard basis vectors in R^n, and I'm wondering what the standard orientations are in n dimensions are and why. For ...
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1answer
62 views

Quaternion isn't normal!

I use the following scheme: Quaternion: [ w, x, y, z ] Then I use Euler Angles and convert it to a quaternion, as following: ...
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29 views

Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
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1answer
52 views

How to deal with multiple representations of quaternions

I'm using a quaternion to represent the orientation in a kalman filter. My algorithm works fine until I rotate "upside down". I think this is because there seems to be multiple ways to represent the ...
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1answer
20 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
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2answers
33 views

Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
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1answer
17 views

How can I transform coordinate systems with quaternions?

I have a coordinate system 0 which I'd first like to rotate about its z-Axis which gives me system 1, and afterwards rotate system 1 about its y-axis which gives me system 2. See picture: Both ...
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115 views

Sources on Hamilton's Discovery of Quaternions

This is a strange question and I'm not sure where to put it; I'm currently writing an essay for a history of maths course, and I've chosen the topic of Hamilton's discovery of the quaternions. I ...
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3answers
126 views

Quaternions: Difference(s) between $\mathbb{H}$ and $Q_8$

What is the difference between $\mathbb{H}$ and $Q_8$? Both are called quaternions.
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1answer
83 views

Quaternion and Matrix

I have a quaternion for rotation and a matrix for changing axis(change coordinate from camera to my rendering scene ). I have tested two method and i except to have equal resuls but results are ...
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1answer
30 views

geometry rotation quaternion

Express the rotation of $\mathbb R^3$ by $\frac{\pi}{4}$ about the $x = y,\ z = 0$ axis by using quaternions and identifying $\mathbb R^3$ with $(i, j, k)$-space. Find the image of the point ...
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1answer
102 views

How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2ln(z_1)}$ and $ln(z_1$) can just be found using: ...
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4answers
77 views

Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
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43 views

Integral elements with predescribed properties in quaternion orders

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
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2answers
19 views

Convert from Quaternary to Hexadecimal

I have the number (23011) in Quaternary and I have to convert it to Hexadecimal. Tried looking up for help online, but other convertors who convert the number without any explanations how it's done ...
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1answer
36 views

non-division quaternion algebra is isomorphic to $2\times 2$ matrices

Let $k$ be a field of characteristic $\neq2$. Let $a,b\in k$ be nonzero elements. Let $A:=\left(\frac{a,b}{k}\right)$ be the quaternion algebra over $k$ with parameters $a,b$. Suppose $A$ is not a ...
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241 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
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Plücker coordinates of the Clifford parallels

Let $$q=\cos\theta+(x_q\textbf{i}+y_q\textbf{j}+z_q\textbf{k})\sin\theta$$ be a unit quaternion parameterised by $\theta\in\mathbb{R}$, where $(x_q,y_q,z_q)$ is fixed and $x_q^2+y_q^2+z_q^2=1$, and ...
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2answers
226 views

Getting Euler (Tait-Bryan) Angles from Quaternion representation

Apologies if this has already been answered, but I haven't been able to get a clear answer from looking on Stack Exchange so-far. I'm trying to solve a camera stabilization problem. I have a 2-axis ...
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1answer
33 views

How can I find a unit velocity vector between two quaternions?

I have two quaternions, $Q_0$ and $Q_1$. I want to find the unit angular velocity vector $w$ that rotates $Q_0$ in the direction of $Q_1$ (shortest path). How can I do this? The analog of what I ...
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1answer
45 views

Preimage of a point by a power map in quaternions

Suppose we have a point $x_0\in{\bf H}$ (where by $\bf H$ I denote the ring of quaternions). What I'm curious about is what can the set of solutions of $x^2=x_0$ look like? From what I've checked, ...
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1answer
58 views

Quaternion derivative w.r.t. its angle

The following quaternion represents a rotation by $\theta$ around the z-axis: \begin{align} q &= (\cos(\frac{1}{2}\theta), \vec{u}\cdot\sin(\frac{1}{2}\theta)), \\ \vec{u}&=(0,0,1)^t ...
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65 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 ...
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1answer
32 views

Quaternions and critically damped spring

I would like to apply critically damped spring smoothing method to smooth movement on the unit sphere to a desired orientation. I have two quaternions, one that represent current orientation and one ...
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1answer
274 views

Quaternion Decomposition

I'm having trouble decomposing a unit quaternion into euler angles (or roll, pitch and yaw). The overall goal is to tell how a phone is rotated with respect to the world. I'm given a unit quaternion ...
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1answer
35 views

About the derivation of a composite quaternion

This problem has been bothering me for several days, hence I decided to ask you for help. I am reading the book "Quaternions and Rotation Sequence" written by Jack B. Kuipers. In section 6.4, the ...
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Dual quaternion derivation

I'd like to derivate a dual quaternion \begin{align} \hat{q}&=(1 + \frac{1}{2}\epsilon\vec{t})q \end{align} where \begin{align} q &= e^\vec{w} , \\\vec{w}&=(0, w_1,w_2,w_3)^t ...
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1answer
48 views

Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...
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1answer
75 views

Unit quaternions as rotations

How would one represent the map $f$ such that $f(1) = i, f(i) = -1$ and keeping $j$ and $k$ fixed as a quaternion representation of rotations?
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20 views

Differentiation and angular velocity of quaternions

I have a list of quaternions that I obtained from a motion capture recording, the sampling rate is 100 Hertz. I need to calculate the regression coefficients for each quaternion, which I can do with ...
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1answer
50 views

smooth orientation change with quaternions

My camera orientation is looking in the $v_1$ direction. Something happens on direction $v_2$ and I want the camera to move smoothly to look at that direction. So, to find the quaternion to go from ...
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17 views

Normalization of Euler angle data

I have head motion data for several speakers. Because not every speaker sat in the exact same position during recording I have to normalize the data. One option to do this, I think, would be to ...
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20 views

What's an hyper-sphere in relation to a quaternion and viceversa?

How you explain in simple words what an hyper-sphere is, assuming that your interlocutor imagines an hyper-cube as a figure composed of 3D cubes in a 4D space ? How you maintain that affinity with the ...
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637 views

Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
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2answers
28 views

How to find angle difference in quatenions?

How does one find the angle difference between two quaternions. There was an answer to this post which says the angle difference between $x$ and $y$ is $z=x\ast \mathrm{conj}(y)$. Is that the ...
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1answer
2k views

Compute Angle Between Quaternions (in Matlab)

I am working on a project where I have many quaternion attitude vectors, and I want to find the 'precision' of these quaternions with respect to each-other. Without being an expert in this type of ...
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3answers
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Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
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0answers
108 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
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3answers
100 views

Quaternions as an Algebra

I'm lacking some vital understanding about quaternions and algebras in general. If we first define $V=\{a+bi+cj+dk|a,b,c,d\in\mathbb{R}\}$. Then we define scalar multiplication, vector multiplication, ...
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51 views

Show by Example that $\mathbb{H}^n$ to $\mathbb{H}^n$ is not necessarily $\mathbb{H}$-linear

Show by example that for$ A \in M_n \mathbb{H}, L_A : \mathbb{H}^n \rightarrow\mathbb{H}^n $ is not necessarily $\mathbb{H}$-linear So I thought it would be linear by definition. Because if we have $ ...
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1answer
3k views

Quaternion distance

I am using quaternions to represent orientation as a rotational offset from a global coordinate frame. Is it correct in thinking that quaternion distance gives a metric that defines the closeness of ...
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1answer
107 views

How to define a quaternion group of order 8

I'm having problems to understand the way this group (Q8) is represented. I have seen definitions using the elements i,j and k, but these same letters don't appear in another definition where each ...
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1answer
51 views

Relative rotations using quaternions [duplicate]

I have a sensor at some arbitrary orientation (non-zero roll, yaw, pitch) given by quaternion $q_{0}$. I then pitch the sensor to orientation $q_{1}$. When I compute the relative rotation between the ...
5
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1answer
72 views

Is there much theory developed for analytic functions of quaternions or of octonions?

The quaternions are associative, so nonnegative integer powers of quaternions are well-defined, and one can consider analytic functions on $\mathbb{H}$ (functions that are given locally by power ...
5
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1answer
333 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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1answer
52 views

Every element with finite conjugates in the ring of real quaternions is a real number

Let $H$ be the ring of real quaternions and let $x$ be a member of $H$ having finite conjugates. Prove that $x$ is a real number. I worked a lot on this question, but no progress! :|