For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.
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votes
0answers
14 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 $$||A-Q||_F =\min_{\Omega \in SO(3)} ...
1
vote
0answers
21 views
“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$
Let a unitary number be one that corresponds to a matrix of the form:
$$\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)$$
This is analogous to ...
4
votes
3answers
146 views
4 dimensional numbers
I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
1
vote
1answer
47 views
Coordinate Transformation on Local coordinate system
I am having a point $P(x,y,z)$ in $3D$ with respect to global coordinate system. I want to create an another Local Coordinate System by picking three points $N1, N2, N3$ in 3D. Now I want to know the ...
5
votes
0answers
40 views
Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$
The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
1
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0answers
23 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
17 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 ...
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votes
0answers
59 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
17 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
17 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 ...
1
vote
3answers
146 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
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 ...
2
votes
2answers
102 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 ...
2
votes
3answers
101 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 ...
