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

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Hurwitz quaternions divisibility problem

Let $\alpha$ and $\beta$ be Hurwitz integral quaternions, and let $p$ be a rational prime such that $p \mid N(\alpha), N(\beta), N(\alpha - \beta)$. Show that $\alpha$ and $\beta$ share a left ...
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1answer
28 views

understanding quaternions - spatial rotations

I would like to know if my understanding about quaternions is correct please: lets say you have a vector in 3d space. You could rotate the x,y and-z frame on a fixed point so that it is parallel with ...
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quaternions - understanding a formula

Quaternions are new for me. I am trying to understand the following formula: What are: $\large{q^x}$ ? I don't think it is a power. $\large{q^t}$ ? just a transposition of the quaternion $q$? ...
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1answer
44 views

what represents this strange matrix [closed]

But I just encountered a very strange type of matrix. What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on ...
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35 views

Rotation quaternions and rotation matrix for electron spin [closed]

Is it possible to construct a rotation quaternion and rotation matrix for the approximation of an electron spin?
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44 views

How do I get the rotation between two rotationmatrices?

I am stuck on a little rotation problem. The problem: I have 2 rotation matrices $A$ and $B$. $A$ and $B$ are relative to the coordinate system O. $A$ and $B$ are Quaternion rotation matrices. I am ...
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1answer
96 views

Eigenvalues of Group Elements and Quaternions

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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1answer
29 views

Question about Eigenvalues of group elements

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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1answer
31 views

What is the mapping from purely imaginary quaternions to a vector in $\mathbb{R}^3$

It is claimed that $q = x{\bf i} + y{\bf j} + z{\bf k}$ has an one to one mapping to a vector $v \in \mathbb{R}^3$ where $v = x \hat i + y \hat j + z \hat k$ But ${\bf i}, {\bf j},{\bf k}$ are ...
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28 views

Prove $SU(2)$ is isomorphic to the group of quaternions of norm 1

How could I start finding the isomorphism? Intuitively, a quaternion can be expressed as two complex numbers $a+bi+cj+dk=a+bi+(c+di)j$, and as an element of $SU(2)$ is $\left[ \begin{array}{ c c } ...
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1answer
58 views

prime divisor propertyfor Hurwitz integers

The Hurwitz integers $\mathcal{H}_{\mathbb{Z}}$ is a particular subset of quaternions. Define: $$ \mathcal{H}_{\mathbb{Z}} = \left\{ a\frac{1+i+j+k}{2}+bi+cj+dk \ | \ a,b,c,d \in \mathbb{Z} \right\} = ...
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14 views

Homomorphisms in $Q_8$ [duplicate]

Prove directly that the 2-dimensional irreducible representation $\rho$ of $Q_8$ is not realisable over $\mathbb{R}$. Suppose $\rho: Q_8 \rightarrow GL_2(\mathbb{R})$ is a representation with ...
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36 views

Rotation that is swapping quaternion $w$ and $x$, $y$ and $z$

I have a quaternion, representing a rotation, equals to: $(w, x, y, z)$ If I convert that quaternion to euler angles, add 180 degrees to $x$ and $z$, and convert it into a quaternion again, I get : ...
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14 views

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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19 views

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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21 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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30 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
13 views

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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0answers
18 views

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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124 views

Minimum number of real multiplications to multiply two quaternions [closed]

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
29 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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2answers
93 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
60 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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2answers
38 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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1answer
47 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
33 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
97 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
65 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
57 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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29 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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2answers
22 views

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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1answer
69 views

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
46 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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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
62 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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1answer
16 views

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
28 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
41 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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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
83 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
28 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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1answer
40 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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2answers
42 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
39 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 ...
2
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0answers
74 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
55 views

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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64 views

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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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 ...