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

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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 ...
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1answer
135 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
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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2answers
3k views

How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?

How would I create a rotation matrix that rotates X by a, Y by b, and Z by c? I need to formulas, unless you're using the ardor3d api's functions/methods. Matrix is set up like this ...
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356 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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1answer
187 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 $SU(2)$ ...
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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
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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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2answers
126 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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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 ...
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1answer
139 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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317 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 ...
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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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1answer
305 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
946 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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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
967 views

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
73 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
42 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 ...
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2answers
157 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
127 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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1answer
42 views

Rotation about an axis by matrix multiplication

Suppose I have three axis of rotation vectors $\vec{v_1},\vec{v_2},\vec{v_3}$ and angle of rotation as vectors $\theta_1,\theta_2,\theta_3$. Take a vector $P$ then apply rotation around $\vec{v_1}$ ...
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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, ...
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121 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 ...
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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 ...