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

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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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44 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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113 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
79 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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67 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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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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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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96 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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47 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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31 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
85 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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22 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
51 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
54 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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51 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
73 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
113 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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31 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
98 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
64 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
51 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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96 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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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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108 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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50 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
53 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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173 views

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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196 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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43 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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38 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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34 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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61 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 ...
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Calculating the perpendicular of a plane using many noisy measurements

I have a number of noisy measurements of quaternions, $q_i$. Where $q_i$ describe rotations from the world frame to a given plane. I want to calculate a perpendicular vector to the plane so as to ...
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1answer
40 views

Align the cube's nearest face to the camera

I have a cube and 4x4 transformation matrix Cube is rotated randomly I need to find the nearest face of cube regarding to camera and rotate the cube by aligning that face to the camera. How can I do ...
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19 views

Eigenvalues of Second Derivatives on Quaternions

So, say you have some quaternion $q = a + b i + c j + d k$. Let $a, b, c, d, \in f(w, x, y, z)$ which each take four real numbers and yield one real number. Then, what are the most general solutions ...
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Is there an algebraic closure for the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r ...
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1answer
131 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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1answer
57 views

Determine similarity between two sequence of quaternions while allowing a degree of freedom around Z axis

A person holds his phone and rotates it in space in a sequence. I am able to obtain a sequence of quaternions from the phone's motion sensors representing the rotation of the phone from the phone ...
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2answers
51 views

Find all Quaternions Satisfying..

Let H be the skew field of quaternions. Find all quaternions x satisfying $(i + j)x(i + k) = 2$ I'm having trouble figuring out what to do with this question. I know the "i j k i j k" formula for ...
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1answer
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Solution of $ax+xb=c$ in a division ring

The equation $ax+xb=c$ in the quaternions skew field ($a,b,c,x \in \mathbb{H}$) has solution: $$ x= \left(|b|^2+2b_0a +a^2\right)^{-1} \left( ac +c \bar b\right) $$ Where $|b|,b_0,\bar b$ are ...
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131 views

Can quaternions be useful for integrals?

Lets assume we want to find a closed form for $\int_0^1 f(x) dx$ where $f(x)$ is a real-analytic function. There are many techniques to find that. Some include contour integration on the complex ...
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1answer
68 views

Isomorphism of Quaternion group

Prove that $Q_{8} \cong H \rtimes G \Rightarrow H=\{e\}$ or $G=\{e\}$. My proof (is it correct?): We know that (it's a fact from the lecture): If $M \cong H\rtimes G$, then $H \cong K \unlhd M$ ...
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72 views

Know if a 4x4 matrix is a composition of rotations and translations (quaternions)

I am using quaternions to describe 3D transformations. A position in space is representated by a (x,y,z,1) vector, and a transformation by a 4x4 matrix, following quaternions logics as far as I could ...
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1answer
93 views

What is the general definition of the conjugate of a multiple-component number?

I know that the conjugate of a binomial is the negation of the second part. So the conjugate of (a + b) would be (a - b). I know that the conjugate of a complex number (a + bi), similarly, is (a - ...