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

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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
344 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 $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: ...
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4answers
123 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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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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0answers
53 views

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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1answer
99 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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1answer
115 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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6k 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
84 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
304 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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129 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
63 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
614 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
166 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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0answers
96 views

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
126 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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1answer
125 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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2answers
242 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
111 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 ...
4
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1answer
60 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
254 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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2answers
54 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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3answers
192 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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1answer
574 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
140 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 ...
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1answer
124 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 ...
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1answer
45 views

Quaternion techniques for a geometric description of the composition of two rotations

Let $q \in S^3$. Therefore $q$ can be represented as $q=\cos(\alpha/2) + \sin(\alpha/2)u$ for some $\alpha \in \mathbb{R}$ and some $u \in S^3$ with it's real part zero. Recall that the quaternions ...
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1answer
333 views

Magnitude of rotation between two quaternions

I have a quaternion for an object's starting rotation, and a quaternion for an object's ending rotation, and I am SLERPing the shortest rotation between the two. How can I figure out the magnitude of ...
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1answer
68 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! :|
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146 views

Relative rotation between quaternions

Say I have a quaternion q which describes how to get from frame 0 to frame 1, and a quaternion r which describes how to get from frame 0 to frame 2. To get the "quaternion difference" between q and r, ...
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334 views

Using quaternions to represent an affine transformation?

I have never used quaternions, so before trying on my problem I would like to know whether this is a good idea: I want to interpolate an affine transformation: I have a set of points in a first 2D ...
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3answers
77 views

Show that for $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1$.

Show that $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1.$ I am pretty sure I can easily google the multiplicative inverse in $\mathbb{H}$, but can you give me a hint on how to ...
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2answers
302 views

Difference between quaternions and rotation matrices

This is a really simple question, I guess. Do quaternions cover the same set of rotations as rotation matrices? I assume the answer is yes, they both can represent SO(3), but I'm unsure about the ...
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1answer
36 views

Enumeration of Hurwitz quaternions of norm p

In "on Quaternions and Octonions" by Conway and Smith, they quote a result by which for each prime norm $p$ there are exactly $p+1$ Hurwitz quaternions of norm $p$. I haven't found any proof of that. ...
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Hamilton's letter to his son

I'm looking for a better reference on this letter from Hamilton to his son where he wrote about his discovering on Quaternions. I'd like to read, if it is possible, a scanned version of the letter. ...
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1answer
48 views

Are there any methods to exponentiate a real number with a number from an arbitrary field?

How can I take the following exponent, for some real-valued number a? $$a^{3+2j-9k+3i}$$ over the field of quaternions, or any field for that matter? On wikipedia we are given the following formula, ...
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1answer
96 views

Why do we start losing algebraic properties when dealing with hypercomplex numbers? [duplicate]

Every form of hypercomplex number I have seen (including the complex numbers) lose some important algebraic property. Why is that? Is there a pattern to what we lose?
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0answers
51 views

How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) ...
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2answers
346 views

Examples of a non commutative division ring

What are some examples of a non commutative division ring other than quaternions?
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1answer
81 views

Why and how are quaternions 'bilinear'?

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as: While others specify: Excuse the poor images, ...
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1answer
85 views

Show that the the multipliction and inverse operations on the quaternion unit sphere are continuous

This is a bit of a tricky question, we define the real Quaternions as: $$H=\left\{ a+bi+cj+dk\mid a,b,c,d\in\mathbb{R}\right\}$$ With the rule that: $$ij=-ji=k\:,\: jk=-kj=i\:,\: ki=-ik=\, j\;,\: ...
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1answer
711 views

Rotation by quaternion conjugation and quaternion matrix

A rotation of vector $v$ can be done by matrix multiplication $Q^{*}Qv$ where $Q=\begin{pmatrix}w & -z & y & x \\ z & w & -x & y \\ -y &x &w& z\\ -x& -y ...
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1answer
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Reference for the polar parameterization of quaternions

I would like to find the original reference in which the polar parameterization of quaternions was given (i.e. the relationship between the components of a unit quaternion and the polar angles of an ...
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1answer
95 views

Automorphism of $Q_8$ [duplicate]

Is there anyone could help me to prove that $Aut(Q_8)=S_4$? Someone told me that there's an isomorphism between the rigid motions of cube and $Aut(Q_8)$, any ideas? Thank you!
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1answer
99 views

Show exponential function maps any line through the origin onto a circle of radius 1 in $\mathbb{S}^3$.

Show that the exponential function maps any line through the origin in $\mathbb{R}$i +$\mathbb{R}$j + $\mathbb{R}$k onto a circle of radius 1 in $\mathbb{S}^3$. I know that for any element v $\in$ ...
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2answers
4k views

How to rotate one vector about another?

Breif Having given 2 non-parallel vectors: a and b, is there any way by which I may rotate a about b such that b acts as the axis about which a is rotating. Question Given: vector a and b To find: ...
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2answers
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Is it possible to use the imaginary components of quaternions to facilitate calculation of vector cross products?

It has come to my attention that the cross products of the vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are almost identical to the products of the imaginary components of quaternions $i$, ...
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1answer
48 views

Set of rotations necessary to connect two points in R³ using a thin cylinder

I have been scratching my head for days trying to answer this question. Suppose i have 2 points on three-dimensional space, say, $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, and they are separated by ...
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0answers
60 views

Present-day uses of quaternions [duplicate]

"Everybody knows" that quaternions are not used for the purposes for which they were originally intended. They are, however, used in computer graphics, and perhaps in astrogation. Besides that, what ...
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292 views

Understanding quaternions & gradient descent in a paper on inertial / magnetic sensor arrays

I hope this question is appropriate here! I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" ...