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

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Quaternion Rotation and Transform Exercises with Answers

I work for a company that develops a lot of navigation based software. Most of the problems that we solve can be tackled using rotation matrices, but I've been doing some reading recently about the ...
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Condition under which Hurwitz quaternion has left or right gcd equal to 1 with its conjugate

Correct me if I am wrong but for Gaussian integer - $a +bi$ its $gcd (a+bi, a-bi) = 1$ - when $gcd (a,b)=1$ and $a +bi \neq 1+i$ I wonder if there are any known conditions under which Hurwitz ...
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15 views

Cartesian extremities of a 3d segment

I have a segment in 3d space and I want to calculate its extremities. I know the cartesian coordinates (x,y,z) of the segment's middle point, the segment's length L and the segment's orientation using ...
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24 views

Grothendieck group of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such ...
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1answer
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Correspondence between rotations and pairs of antipodal unit quaternions

I'm having some trouble understanding how rotations of $\mathbb{R}^3$ correspond to antipodal pairs of unit quaternions. In section 1.5 of his Naive Lie Theory, John Stillwell proves the theorem that ...
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How can I transform coordinate systems based on quaternion data?

I have a single rigid body object, and its orientations in quaternion with respect to two coordinate systems, each is called original and prime, respectively; therefore, I have two quaternions ...
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79 views

Minimum number of real multiplications to multiply two quaternions

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
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1answer
28 views

Showing two definitions of the Quaternion Algebra are the same

For $q=z+jw$ where $z,w\in\mathbb{C}$, I'm given a map $M:\mathbb{C}^2\rightarrow M_{2\times2}(\mathbb{C})$ given by $$M(q)=\begin{pmatrix} z & \overline{w} \\ -w & \overline{z} ...
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90 views

How are quaternions a finite set?

I'm having trouble understanding how Quaternions are a finite set when you can express a quaternion as Q = a + ib + jc+ kd, because a, b, c, d are $\in$ of $\Re$ would this not mean that the set is ...
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1answer
55 views

Can closure of quaternions under multiplication be shown with a cayley table?

Unsure about my understanding of groups and quaternions. I'm trying to figure out if just using a cayley table (specifically this one) can show closure of quaternions under multiplication, is ...
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32 views

Quaternion order associated to a ternary quadratic form

I am a bit puzzled by the discriminant of a ternary quadratic form. According to Lehman 1992 and another related question, the discriminant of a ternary quadratic form is the half-determinant of its ...
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45 views

Question about $4\times4$ matrix representation of a quaternion

I have a problem to solve about showing the real quaternion group $\mathbb{H}$ is isomorphic to $M_4(\mathbb{R})$ When trying to define my map I was having trouble coming up with an appropriate map ...
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1answer
28 views

3 rotation values to work out rotation in degrees

I am currently working with the Oculus headset and dealing with the Z axis. With the software I have, the values I can retrieve are limited and I was hoping someone could help me find a solution to ...
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2answers
92 views

Quaternion ^ Quaternion [duplicate]

I was looking at Quaternions at Wikipedia - I was trying to find the value of $i^j$ etc... Wikipedia lists $q^\alpha$ where $\alpha$ is real, but I can't find the value of $i^j$. Any clues? The ...
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2answers
61 views

How would I apply an Exponential Moving Average to Quaternions?

I'm trying to filter positional and rotational data using an Exponential Moving Average (EMA) filter. This has worked without issues for positional data (3D vectors) but I can't figure it out for ...
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2answers
51 views

Quaternion algebra of characteristic 2?

I've been reading up on quaternion algebras recently and noticed the vast majority of theorems are contingent on setting the characteristic $p \neq 2$. In particular, this is true for the central ...
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26 views

Euler -> Quat: Flipped: 90 == -270

i am playing around with quaterions, matrices, euler rotations. For some reason when converting from euler to quaternion to euler my angles are flipped. So where i expect 90, i get -270. 30 stays ...
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Commutativity of Spatial Rotations

I know that in general spatial rotations (rotations in $\Bbb R^3$) are not commutative. But what if we restricted our possible rotations to only those around orthogonal axes? For instance, what if ...
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What is the difference between Quaternions and Bicomplex Numbers?

So, I know Quaternions are basically 4 dimensional Complex numbers, and the dimensions can double forever to Octonions, Sedinions, etc. I recently heard about bicomplex numbers, which are also sort of ...
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1answer
45 views

Where is a good starting place to do research on the algebra of quaternion numbers?

I'm doing a project for my intro to real analysis class and decided that the algebra of quaternion numbers would be interesting to do. I'm wondering what a good starting place would be.
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21 views

Trouble with using dual quaternions to control the translation and rotation of objects

Okay so I'm using dual quaternions to control rotation and translation of a skeleton and I've encountered a problem. The problem probably has an incredibly simple solution but the problem itself is ...
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1answer
48 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 ...
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26 views

Division algebra over 2-adic fields

Let $D$ be the quaternion division algebra and $O$ be a maximal $\mathbb{Z}$-order in $D$, say the Hurwitz quaternion integers. It can be proved that $D$ and $O$ split at odd primes, that is ...
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Fourier-transform of fermionic model: book recommendation

I am currently interested in the mathematics of fermionic fock spaces and especially its Fourier-transform $$ \mathcal F(g)(y)=\int \exp \left(-\sum_{i=1}^n y_ix_i\right)\cdot g(x) dx $$ with the ...
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1answer
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2d indicator for turning a spacecraft in 3d space

For the admins Please look at the tags.... I have no idea where to put this in math I also posted this here http://www.gamedev.net/topic/666267-2d-indicator-for-turning-a-spacecraft-in-3d-space/ ...
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1answer
23 views

Rotate the segment by quaternion - how to find actual segment's end position?

I have an segment from [0,0,0] to [0,1,0] (left-handed coordinate system, with Y axis up) which is non-rotated. The rotation is ...
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1answer
35 views

Solving a transformation equation involving vectors and quaternions

I'd like to solve the following equation for $c$, where $a$, $c$, and $d$ are position vectors represented by quaternions with $w$ (the real component) set to $0$ and $b$ is a unit quaternion: ...
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38 views

Clifford algebra and Spin group of 4-dimensional Euclidean space

I’m seeking for a straightforward construction of well-known $\mathrm{Spin}(4) = \mathrm{Spin}(3)\times\mathrm{Spin}(3)$ isomorphism using geometric algebra-based definition of “Spin”, without ...
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2answers
65 views

How can one intuit complex numbers from quaternions?

I understand that quaternions are sort of an extension of complex numbers in higher dimensions. If that's really the case conceptually (is it?), it must be possible to get back from the higher ...
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1answer
72 views

Vector spaces over quaternions

Let $V$ be an $n$-dimensional vector space over the quaternions $\mathbb{H}$, and let $G$ be the multiplicative quaternion group. How would one show that $V$ would then be a $4n$-dimensional ...
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1answer
23 views

THE positive half-spin space of quaternion vector space

I have the following information: $T$ is the one-dimensional quaternion vector space with the canonical action of $\Gamma$, a finite subgroup of SU$(2)$. This makes sense as SU$(2)$ is the unit ...
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28 views

How to find rotation quaternion for a model so that it is perpendicular to a line in 3D space?

How to find the target rotation quaternion for a model when one of its faces need to be aligned perpendicular to a line in 3D space. For example, if the model is a cube and if two 3D points connecting ...
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31 views

Handedness in Quaternion multiplication

I've received some code (which I didn't write) and decided at some point to write test cases for the Quaternion math implementation. I used Wolfram alpha to get the result q1 * q2, where: q1 = (4.0 ...
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1answer
34 views

Rotate a vector about a given axis by the use of a quaternion

I encountered a problem in programming where I need to rotate a given vector about a given angle. To be precise, I need to change it to a quaternion so that I can later change it to a 4x4 matrix to ...
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59 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
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2answers
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How is it possible that a single Quaternion can be expanded into it's forward, up and right vector components?

So, I've been writing this raytracer for my own entertainment and recently I discovered Quaternions so I decided to implement a camera that uses them. After a lot of struggle I finally managed to ...
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Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
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44 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 ...
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45 views

Quaternion relation proofs (e.g.: $ik=-j$)

How do you prove that these relations are correct $(ij = k, jk = i, \ldots)$? I tried to prove some of them, and I could, but for example: ...
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Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
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1answer
31 views

Rotating a point in space about another via quaternion

I have a system that is giving me a point in 3D space (call it (x, y, z)) and a quaternion (call it (qw, qx, qy, qz)). I want to create a point at (x+1, y, z), and then rotate that point using the ...
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1answer
43 views

Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
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Alternative quaternion multiplication method

Given two quaternions, $a+bi+cj+dk$ and $e+fi+gj+hk$, their product (w.r.t. their given order) would normally be given by ...
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1answer
102 views

Converting Quaternion or 4x4 Matrix to 3x3 Matrix representation.

I'm working on some code that manipulates an Axis-Aligned Bounding Box, so it always encompasses the object it borders. I use a 3x3 matrix to re-size the box when it rotates. The only issue is I only ...
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Suggestions for Clifford Analysis

I'm trying to learn Clifford Calculus and I can't find introductory level books. I found some nice lecture notes of José Figueroa-O'Farrill when I was studying Clifford Algebra: ...
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What is the right name for the space occupied by a quaternion

I have a little problem wrapping my head around quaternions, in particular I have problems about how to pair the usual "3D algebra" with the theoretic vision of a quaternion. I know that informally a ...
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41 views

Conjugation of quaternions

This proof is extreeeeemely boring, but still I must get it right. Let $x = x_0 + x_1 i + x_2 j + x_3 k \in \mathbb{H}$ (the Hamilton quaternions). Conjugation is defined as: $$x^\ast = x_0 - x_1 i - ...
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1answer
19 views

Velocity vector transformations with respect to a global frame of reference

This seems like it should be a simple problem, but I've been stuck on it for about a day now. It's technically a programming problem, but I'm posting it here because the root of the problem is really ...
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22 views

Tait–Bryan angle rotations using quaternions

I am attempting to rotate a point $P$ in $\mathbb R^3$ using the Tait–Bryan angles as the inputs, but the math is done by quaternions. This is what I have done so far: Let the Tait-Bryan angles be ...
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1answer
42 views

The major differences between dual-quaternions and screw theory

I am working on a project involving the motion of rigid body. From the literatures, I found two main tools, namely the dual-quaternions and screw theory. May I ask what are the major differences ...