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

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208 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
4
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1answer
324 views

Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations $S^1\to\...
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2answers
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Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
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1answer
71 views

Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
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1answer
37 views

What does it mean that quaternions/ spinors are negated under a full rotation?

As I understand it, quaternions are a type of object called a spinor. Spinors are objects that are negated under a full rotation and only return to their original state under two full rotations. But ...
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3answers
503 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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2answers
301 views

Quaternions as roots

So, I StumbledUpon this really cool site and the last picture looked almost as if it had 3D structure. This reminded me of another website where I saw pictures of the order-8 Mandelbulb. I got to ...
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0answers
200 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 ...
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1answer
1k views

Interpolating Rotation Quaternions

Suppose I've got two quaternions that each represent an angle. I need to interpolate between these two angles (from 0% to one side to 100% to another side). Since I work a lot with complex numbers, I'...
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2answers
77 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 ...
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1answer
7k views

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
249 views

Quaternions as a Lie algebra, its derivations

Let $\mathbb{H}$ be the algebra of quaternions. It can be proven that each derivation $D:\mathbb{H}\to \mathbb{H}$ is inner that is of the form $\mathrm{ad}x$ for some $x\in \mathbb{H}$. I am to prove ...
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2answers
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Concise description of why rotation quaternions use half the angle

I'm currently writing the report on my master thesis project, where I use Android sensors to perform inertial navigation in a heavy industrial environment. In my application, I make use of quaternions ...
2
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1answer
345 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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4answers
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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 ...
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1answer
1k views

Rotation by quaternion conjugation and quaternion matrix

A rotation of vector $v$ can be done by matrix multiplication $Q^{*}Qv$ where $Q=\begin{pmatrix}w & -z & y & x \\ z & w & -x & y \\ -y &x &w& z\\ -x& -y &...
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2answers
282 views

How can i express a quaternion in polar form?

Im trying to express the following quaternion in polar form (axis-angle) $a=1+i-2j+k$ Would this be the resultant ? $$\cos \frac{θ}{2} +\sin \frac{θ}{2} (i-2j+k)$$
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1answer
186 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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1answer
380 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
2
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1answer
151 views

Quaternion exponential

Given an imaginary quaternion $ \mathbf{v}=\alpha \mathbf{i}+ \beta \mathbf{j}+\gamma \mathbf{k} $ its exponential is: $ e^\mathbf{v}=\cos \theta +\mathbf{v}\dfrac {\sin \theta}{\theta} $ where $\...
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1answer
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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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1answer
633 views

Quaternion - Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). And I am trying to calculate the angle of rotation around all the three axes. I have tried may methods but not getting the ...
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0answers
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Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship $...
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1answer
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How can I transform coordinate systems with quaternions?

I have a coordinate system $0$ which I'd first like to rotate about its $z$-axis which gives me system $1$, and afterwards rotate system $1$ about its $y$-axis which gives me system $2$. See picture: ...
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1answer
76 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
58 views

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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1answer
105 views

Quaternion to Euler with some properties

I am trying to create a map editor (for GTA SA-MP), and the source game data contains objects with quaternion rotation, whereas I need the editor to output the objects with Euler rotation (XYZ) in ...
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1answer
2k views

Quaternion and rotation about an origin and an arbitrary axis origin help

Greetings All Thanks to James and Chas for helping me get this far btw Chas the language I wrote it in is in matlab. I tried to respond to your post but wasn't able to do it..I guess the gremlins ...
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2answers
209 views

Rotation Equivalence using Quaternions

I'm given a statement to prove: A rotation of π/2 around the z-axis, followed by a rotation of π/2 around the x-axis = A rotation of 2π/3 around (1,1,1) Where z-axis is the unit vector (0,0,1) ...
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1answer
141 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
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29 views

How to calculate rotation quaternion between two orientation quaternions?

I have some device (3D pointer) connected to my computer which returns it's position (in cartesian XYZ system) and orientation (in quaternions). I receive this values about 30 times/sec. Now I need ...
0
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1answer
215 views

Relative rotation between quaternions

Say I have a quaternion q which describes how to get from frame 0 to frame 1, and a quaternion r which describes how to get from frame 0 to frame 2. To get the "quaternion difference" between q and r, ...
0
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1answer
48 views

Can quaternions be used to represent rotation rate?

A quaternion is a useful tool for representing a rotation, or change in attitude. If a quaternion $q$ represents a rotation, and $v$ a vector, then $v'=qvq^*$ rotates the vector, where the multiply ...
0
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1answer
49 views

Equivalance form for Slerp in quaternions interpolation

In all the books I have found that Slerp have two forms: A B I know that all the forms from A are equivalent but I don't know why the forms from A are equivalent with the form from B. Can ...