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

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Quaternions, torque, and impulse.

In a physics simulation I have a solid ball of mass $m$ and moment of interia $M$ (which is a diagonal matrix with all entries equal to ${2\over5}mr^2=i$). Its instantaneous rotation is given by a ...
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3answers
590 views

half sine and half cosine quaternions

Something is a little bit unclear to me. In the image below you see that you need to divide the angle by a half. Acccording to wikipedia they say that this is so that I could rotate clockwise or ...
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4answers
1k views

Quaternions: why does ijk = -1 and ij=k and -ji=k

Currently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: ...
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2answers
321 views

Detail in the proof of the quaternion rotation identity

I am trying to understand the proof of the quaternion rotation identity illustrated in wikipedia ...
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2answers
215 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
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1answer
276 views

Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
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1answer
83 views

How can I break down a rotation of known amount around a known axis into two rotations of unknown amounts around known axes?

I have a vector that's been rotated a known amount about a known axis. I would like to break this rotation down into two separate rotations around known, linearly independent axes where the amounts I ...
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1answer
140 views

How to Perform Quaternion Multiplication

Everywhere that I've looked, it seems to be assumed that $i^{2} = j^{2} = k^{2} = - 1$, along with the other rules of quaternion multiplication. However - for my homework - I'm being asked to show ...
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1answer
105 views

inner product on matrices with quaternionic entries

let $\mathbb{H}$ be the quaternions and $Mat(n,\mathbb{H})$ be the vector space of $n\times n$ matrices over $\mathbb{H}$. Let $H(n,\mathbb{H}):=\{ A\in Mat(n,\mathbb{H}): \overline{A}^t=A \}$ be the ...
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1answer
521 views

3 axis gimbal controller and quaternions

this question has been probably asked in different forms but please bare with me: I'm building a three axis gimbal controller as part of my uni project. Besides the gimbal stabilization on each axis, ...
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2answers
547 views

Updating a quaternion orientation by a vector of euler angles

I'm trying to understand why this formula works to update an orientation with an angular velocity represented as a vector of rotations in $radians/{second}$. I understand that two quaternions ...
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1answer
180 views

how to extrapolate quaternions?

from http://answers.unity3d.com/questions/168779/extrapolating-quaternion-rotation.html ...
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1answer
78 views

Error in Weyl character formula computation.

I need someone with a keen eye for errors. I am trying to use the Weyl character formula for the symplectic group Sp$(4,\mathbb{C})$ on certain matrices coming from 2x2 quaternion matrices. Summing ...
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2answers
760 views

What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
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1answer
78 views

Sub division rings of division rings

Below $^\ast$ denotes "nonzero elements of". There is a problem in Jacobson's Basic Algebra 1, there is a problem to this effect: if $S$ is a subdivision ring of $\mathbb{H}$ such that $S^\ast$ is a ...
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1answer
158 views

Triple quaternion multiplication

I'm self learner and for some reason I can't wrap my head around quaternion multiplication. I just stumble upon one of equation in my text. Can anyone show step-by-step workout for below: $$ ...
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3answers
161 views

How can we make $\mathbb{R}^n$ into a multiplicative group?

Are there any sorts of multiplicative group structures we can put on set $\mathbb{R}^n$? By multiplicative, I mean that it should be compatible with the natural scalar multiplication on ...
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1answer
69 views

How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from…

In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$). ...
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1answer
438 views

3D points rotation to quaternions

For the simplicity, we'll consider two 3D points, that moves one relatively to other, in time. Let's say: ...
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1answer
51 views

quaternion product distributivity

If you check the quaternion product derivation at wikipedia: http://en.wikipedia.org/wiki/Quaternion#Hamilton_product You can see that it is derived from a multiplication table between the ...
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1answer
387 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
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1answer
254 views

Why do Quaternions and octonions exist?

Ok so I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are ...
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2answers
100 views

Correspondence between rotation representations

I was wondering if there is a bijection between unit quaternions and other rotation representations such as vector of rotation, Euler angles or rotation matrices. It seems to me this is not the case ...
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2answers
447 views

Quaternion Differentiation

I have an application that tracks an image and estimates its position and orientation. The orientation is given by a quaternion, and it is modified by an angular velocity every frame. To predict the ...
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0answers
150 views

how to fine the quaternion between two vectors lies on two different Cartesian coordinate systems

I have two vectors lies on different cartesian coordinate systems. I want to find the quaternion between these two vectors. how I can do that?
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1answer
342 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
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1answer
270 views

how to get rotation component of quaternion form using 3d coordinates

I have a series of 3d coordinates distributed in a 3d space according to a root point. I can determine the x, y , z component using reducing the vectors. but I am not clear how to get rotation ...
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8answers
3k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
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1answer
261 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
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1answer
429 views

Quaternions and Rotations

Two of the interesting achievements in Mathematics are Classification of platonic solids, and also classification of finite groups acting on the unit sphere in $\mathbb{R}^3$, and they are very nicely ...
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0answers
205 views

Quaternions in olympiad 3d geometry

It's known that we can use complex numbers to solve some 2d problems easier than synthetic methods. But, what do you think about using complex numbers in 3d geometry? I've found extend of complex ...
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1answer
89 views

Why is the square of quaternions not half of all axes angles expressed by this quaternion?

I want to devide a rotation, which is expressed as a quaternion. So I am doing it with Quaternion^POWER, where power is lower than 0. See my question before: here If I calculate following example ...
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2answers
141 views

Why does $i$ have infinitely many conjugates in $\mathbb{H}$?

Browsing this question: Why are the solutions of polynomial equations so unconstrained over the quaternions?, the pdf linked in the comments says that the infinitely many conjugates of $i$ in ...
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1answer
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Compute Angle Between Quaternions (in Matlab)

I am working on a project where I have many quaternion attitude vectors, and I want to find the 'precision' of these quaternions with respect to each-other. Without being an expert in this type of ...
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1answer
204 views

Unit elements in Hurwitz quaternions

Hurwitz quaternions are defined as: $$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$ (that is, all integer or half integer quaternions). ...
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1answer
129 views

Left and Right Vector bundles

I am reading a paper that starts talking about 'left vector bundles' and I'm having trouble figuring out what they mean. The specific setup is as follows: A quarternionic line bundle $L$ over ...
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3answers
593 views

How to get a part of a quaternion? e.g. get half of the rotation of a quaternion?

if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation? Thanks in advance!
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1answer
354 views

Splitting of quaternion algebras

A rational (definite) quaternion algebra is an algebra of the form $$ \mathcal{K} = \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha \beta $$ with $\alpha^2,\beta^2 \in ...
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0answers
355 views

How to convert Yaw, Pitch, Roll and Acceleration value to cartesian system?

I am having readings of Yaw, pitch, Roll, Rotation matrix, Quaternion and Acceleration. These reading are taken with frequency of 20 (per second). They are collected from the mobile device which is ...
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2answers
150 views

Why is any proper division subring of $\mathbb{H}$ contained in the center $Z(\mathbb{H})$?

Here is an idea I've been working on for self study. Suppose $S$ is a division subring of $\mathbb{H}$ (the quaternions, viewed as a subring of $M_2(\mathbb{C})$), which is stabilized by the maps ...
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1answer
58 views

Why do $\mathbb{C}$ and $\mathbb{H}$ generate all of $M_2(\mathbb{C})$?

For this question, I'm identifying the quaternions $\mathbb{H}$ as a subring of $M_2(\mathbb{C})$, so I view them as the set of matrices of form $$ \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} ...
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1answer
354 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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1answer
205 views

Mapping from a rotational quaternion to a single angle

I have given an attitude quaternion that describes the rotation of a device. This roation is relative to the magnetic north, so the quaternion includes implicit information about the heading of the ...
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1answer
528 views

Adding angular velocity values

I'm creating a game in which the user directly controls the tilt of a platform. Since most of the time we will be balancing and controlling a ball, this means large angles and full rotational ...
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2answers
1k views

associativity of quaternion multiplication

I understand that quaternion multiplication is non-commutative, but what association does it have. putting into context when we have the statement when given general numbers, and algebra such as: ...
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1answer
480 views

converting quaternion to vector3

I am considering taking a given rotation quaternion $Q$, about a given point $P1$, and obtaining a vector3 $V$. my thought pattern is thus: given a point $P1$, and a quaternion $Q$ obtain another ...
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0answers
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A question about hypercomplex numbers: quaternions, octonions etc [duplicate]

Possible Duplicate: Why is 8 so special? First of all let me state that I am not a mathematician but I work in computer science and engineering. I was reading about hypercomplex numbers, ...
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1answer
205 views

Euler angles, quaternions and hyperspace

In three-dimensional space it's possible to define rotations using the Euler angles $(\Psi,\Theta,\Phi)$ or quaternions $(i,j,k)$ If we have a hyperspace with more than three coordinates, is it still ...
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1answer
810 views

Uniform Random Quaternion In a restricted angle range

I'm trying to sample uniform random rotations. I'd like the rotations to be restricted in a range [-θ,θ]. I found a method by K. Shoemake which can be summarized as: ...
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4answers
2k views

Quaternions and spatial translations

From my understanding, in spatial applications (3D rendering, games and similar applications) quaternions can only be used to describe rotations/orientations and not translations (like a ...