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

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6
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3answers
290 views

4 dimensional numbers

I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
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1answer
906 views

Coordinate Transformation on Local coordinate system

I am having a point $P(x,y,z)$ in $3D$ with respect to global coordinate system. I want to create an another Local Coordinate System by picking three points $N1, N2, N3$ in 3D. Now I want to know the ...
9
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2answers
400 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
3
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0answers
162 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
2
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0answers
161 views

Understanding quaternions and axis angle representations

I have a sensor that gives me a quaternion. I convert the quaternion to an axis-angle representation using http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToAngle/. When I ...
1
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3answers
3k views

What does multiplication of two quaternions give?

I'm using quaternions as a means to rotate an object in the application I'm developing. If one quaternion represents a rotation and the second quaternion represents another rotation, what does their ...
0
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1answer
102 views

Commutative applying rotations around three axis

Rotating an object in a 3 dimensional space by euler angles might be intuitive but comes with some problems. First, the order of applied rotations around the different axis matters. Second, there is ...
2
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2answers
89 views

Equivalent conditions of quaternion matrix algebra

I am following Theorem 2.3.1 of Maclachlan's and Reid's The Arithmetic of Hyperbolic 3-Manifolds. We define a quaternion algebra $A=\left(\frac{a,b}{F}\right)$ over a field $F$ of characteristic ...
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3answers
3k views

Nullify (zero out, cancel) rotation in an arbitrary axis in a Quaternion

Question: How do you nullify (zero out) rotation around an arbitrary axis in a Quaternion? Example: Let's say you have an object with quaternion orientation $A$. You also have a rotation quaternion ...
3
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1answer
79 views

Some questions about quaternions.

It is possible make something like complexification of a real vector space using quaternions? If yes, it's similar to complex case or there are considerable differences? Has been studied a quaternion ...
2
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2answers
485 views

Proof that Quaternion Algebras are simple

I have a proof that every quaternion algebra over a field $A=\left(\frac{a,b}{F}\right)$ is simple, i.e. has no nontrivial two-sided ideals, which appeals to the algebraic closure of $F$ and the ...
3
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3answers
235 views

Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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0answers
54 views

Optimimal rotation using non-linear conjugate gradient

The problem I'd like to ask is the following : let $M_1$ and $M_2$ two rigid bodies with a quadratic constraint function $f$ attached to its grid points. $M_2$ is always kept static while $M_1$ can be ...
4
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2answers
6k views

Combining rotation quaternions

If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis. The the order of rotation ...
6
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2answers
1k views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
0
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1answer
38 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
4
votes
3answers
496 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
3
votes
1answer
331 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
3
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2answers
65 views

What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ...
1
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1answer
57 views

How does one derive this rotation quaternion formula?

given an angle and an axis, the corresponding quaternion can be computed like this. $w = \cos( Angle/2)$ $x = \text{axis}.x * \sin( Angle/2 )$ $y = \text{axis}.y * \sin( Angle/2 )$ $z = ...
1
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0answers
231 views

How is the Quaternion multiplication derived?

Quaternion multiplication seems suspiciously similar to the cross product. How is it derived? Here is a description of the multiplication: Let $Q_1$ and $Q_2$ be two quaternions, which are defined, ...
3
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3answers
139 views

Quaternions: Difference(s) between $\mathbb{H}$ and $Q_8$

What is the difference between $\mathbb{H}$ and $Q_8$? Both are called quaternions.
0
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1answer
467 views

How do I calculate a position in front of a quaternion given the initial position and the quaternion?

Well I want to determine the position in front of, lets say 'an object'. 'An objects' has a position (pX,pY,pZ) and a quaternion rotation (qX,qY,qZ,qW) (where qW seems always to be ...
0
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1answer
225 views

How are these formulas for Quaternion -> Rotation Matrix related?

I'm trying to write a program to convert a quaternion to a rotation matrix. One source I found is: ...
1
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1answer
153 views

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 ...
1
vote
3answers
940 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 ...
13
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4answers
2k 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: ...
2
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2answers
463 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 ...
7
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2answers
307 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 ...
3
votes
1answer
716 views

Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
2
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1answer
108 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 ...
4
votes
1answer
161 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 ...
1
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1answer
125 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 ...
2
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1answer
734 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, ...
1
vote
2answers
778 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 ...
1
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1answer
288 views

how to extrapolate quaternions?

from http://answers.unity3d.com/questions/168779/extrapolating-quaternion-rotation.html ...
1
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1answer
97 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 ...
23
votes
2answers
822 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 ...
2
votes
2answers
91 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 ...
1
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1answer
180 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: $$ ...
1
vote
3answers
165 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 ...
1
vote
1answer
70 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 $\color{blue}{blue}$). ...
0
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1answer
543 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
68 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 ...
3
votes
1answer
650 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 ...
6
votes
1answer
375 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 ...
3
votes
2answers
125 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 ...
2
votes
2answers
1k 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 ...
4
votes
1answer
448 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
359 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 ...