For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.
9
votes
1answer
87 views
Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?
Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...
0
votes
0answers
36 views
Quaternion exponential map, rotations and interpolation
A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
2
votes
2answers
20 views
Rotate the axis of rotation of a quaternion by another quaternion
I have two quaternions, $q_1$ and $q_2$. I want to rotate the axis of rotation of $q_1$ by $q_2$. Is there any way of doing this directly, without extracting the axis of rotation from $q_1$, rotating ...
0
votes
0answers
16 views
remove rotation about Z-Axis from Quaternion
I have a quaternion representing the rotation of a uav.
The coordinate system is with X/Y defining the horizontal plane and the Z-Axis going up.
For the motor controller i need the rotation WITHOUT ...
2
votes
3answers
56 views
Are there different conventions for representing rotations as quaternions?
I am trying to understand how quaternions are represented as rotations, in particular how to convert from a quaternion representation to a rotation matrix. The following paper by Diebel gives an ...
0
votes
3answers
50 views
Composition of two axis-angle rotations
Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
4
votes
2answers
84 views
Does the square root of $i$ necessitate quaternions?
The square root of i is $\frac{\sqrt{2} + i \sqrt{2}}{2}$. But how is it valid to use a number in expressing the square root of that number?
1
vote
1answer
95 views
Orthogonal procrustes problem using quaternions
Hello I'm trying solve orthogonal procrustes problem in 3d with a help of quaternions.
Original problem is:
For matrix $A$ find orthogonal matrix $Q$ that $$||A-Q||_F =\min_{\Omega \in SO(3)} ...
6
votes
3answers
174 views
4 dimensional numbers
I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
1
vote
1answer
79 views
Coordinate Transformation on Local coordinate system
I am having a point $P(x,y,z)$ in $3D$ with respect to global coordinate system. I want to create an another Local Coordinate System by picking three points $N1, N2, N3$ in 3D. Now I want to know the ...
1
vote
0answers
23 views
“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$
Let a unitary number be one that corresponds to a matrix of the form:
$$\left(
\begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x)
\end{array}
\right)$$
This is analogous to ...
1
vote
1answer
66 views
How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from…
In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$).
...
1
vote
3answers
195 views
Nullify (zero out, cancel) rotation in an arbitrary axis in a Quaternion
Question:
How do you nullify (zero out) rotation around an arbitrary axis in a Quaternion?
Example:
Let's say you have an object with quaternion orientation $A$.
You also have a rotation quaternion ...
6
votes
0answers
55 views
Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$
The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
7
votes
1answer
183 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 ...
2
votes
2answers
177 views
Does quaternion multiplication relate to Minkowski space?
A quaternion notated as $a+bi+cj+dk$ can also be written in terms of a scalar and a vector $(a,v)$, where $v$ is the three-vector $(b,c,d)$. In this notation, the real part of the product $(p,q)(r,s)$ ...
1
vote
0answers
30 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 ...
3
votes
0answers
27 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 ...
0
votes
0answers
63 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
votes
1answer
18 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 ...
1
vote
1answer
18 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 ...
2
votes
2answers
112 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
votes
1answer
57 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 ...
5
votes
1answer
116 views
Why are properties lost in the the Cayley-Dickson construction?
Motivating question: What lies beyond the Sedenions?
I'm aware that one can construct a hierarchy of number systems via the Cayley-Dickson process:
$$\mathbb{R} \subset \mathbb{C} \subset ...
13
votes
1answer
358 views
What lies beyond the Sedenions
In the construction of types of numbers, we have the following sequence:
$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$
or:
$$2^0 \mathrm{-ions} \subset ...
2
votes
3answers
103 views
Math beyond Quaternions
Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
1
vote
0answers
24 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 ...
3
votes
2answers
45 views
Combing 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 ...
3
votes
1answer
65 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
votes
1answer
29 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
89 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
80 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 ...
1
vote
1answer
32 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 = ...
3
votes
2answers
41 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
vote
0answers
54 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
votes
3answers
101 views
Quaternions: Difference(s) between $\mathbb{H}$ and $Q_8$
What is the difference between $\mathbb{H}$ and $Q_8$? Both are called quaternions.
1
vote
1answer
42 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 ...
1
vote
1answer
73 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 ...
0
votes
1answer
47 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
vote
1answer
142 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 ...
8
votes
4answers
291 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
votes
2answers
85 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 ...
4
votes
2answers
104 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 ...
2
votes
1answer
97 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
votes
1answer
46 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
111 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
vote
1answer
46 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
votes
1answer
171 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, ...
3
votes
2answers
96 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 ...
4
votes
2answers
257 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 ...
