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

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2
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1answer
161 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 ...
0
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2answers
448 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 ...
0
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1answer
127 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
67 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. ...
10
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1answer
320 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
57 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 $ ...
10
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3answers
223 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
1k 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 ...
0
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1answer
191 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
153 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 ...
1
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1answer
47 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
404 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
70 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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0answers
180 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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2answers
409 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 ...
5
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3answers
81 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
380 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 ...
2
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1answer
40 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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0answers
60 views

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. ...
3
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1answer
49 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
141 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
59 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
571 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
100 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
92 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\;,\: ...
2
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1answer
997 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 ...
0
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1answer
28 views

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 ...
0
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1answer
109 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!
1
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1answer
109 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 ...
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3answers
5k 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: ...
7
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2answers
187 views

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
55 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 ...
1
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1answer
374 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" ...
4
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1answer
187 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
6
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2answers
222 views

Why is $x_1 i = i x_1$ for quaternions?

According to Wikipedia, $$x+y = (x_0+y_0)+(x_1+y_1) i+(x_2+y_2) j+(x_3+y_3) k$$ and $$\begin{align} x y &=( x_0 y_0 - x_1 y_1 - x_2 y_2 - x_3 y_3)\\ &+( x_0 y_1 + x_1 y_0 + x_2 y_3 - x_3 ...
6
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2answers
226 views

Why is $(-1) \cdot j = j \cdot (-1)$ for quaternions?

I'm currently trying to understand the following part of a script (translated from German to English). It is the first part where quaternions get introduced, so I don't know anything about them except ...
4
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2answers
364 views

Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions

There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions ...
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1answer
475 views

Small angular displacements with a quaternion representation

I have the orientation of a 3D spatial object represented by a unit quaternion: $$ q = a_1 + a_2 i + a_3 j + a_4 k $$ $$ \|q\| = (a_1^2 + a_2^2 + a_3^2 + a_4^2)^{1/2} = 1 $$ I'd like to perturb this ...
4
votes
2answers
423 views

Learning track for quaternions?

I've been out of the math loop for the last decade (although I'm a programmer, I've not done anything with calculus or above for this long) and I'd like to learn about quaternions (particularly ...
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1answer
88 views

Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
0
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1answer
582 views

3D Positioning Vectors to Matrix & Quaternions

This is also programming related, but I think it's more about math than anything else. I have an Object which I want to represent it's 3d position, rotation and scale with vectors, and then I need to ...
4
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1answer
131 views

Mean value of the rotation angle is 126.5°

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
2
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1answer
485 views

Getting cumulative Euler angle from a single Quaternion

I've got an app that uses quaternions, and I'd like to convert each quaternion to the corresponding Euler angles. The issue is, when I convert them, the roll and yaw are bounded within 360 degrees ...
12
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1answer
532 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
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0answers
163 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
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2answers
348 views

Sylow 2-Groups of a Special Linear Group

Let $SL_2(\mathbb{F}_3)$ be the special linear group over the finite field $\mathbb{F}_3$. Show that any Sylow 2-group of $SL_2(\mathbb{F}_3)$ is isomorphic to the quaternion group of order 8.
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2answers
74 views

Immersion of Quaternions

Does there exist an immersion of the Quaternion Group in the Symmetric Groups $S_6$ and $S_7$? If it does exist, can you give me an explicit description of that immersion?
6
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3answers
1k views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
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0answers
65 views

Quaternion Hermitian diagonalization

How do I go about diagonalizing such a matrix. For example let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11$ and take the matrix: $A = \begin{pmatrix} 11 & -j-3k \\ ...