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

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What is the perspective projection of a 3d point relative to a quarternion encoded camera?

I'm representing a camera on the cartesian space as a tuple of a 3d point (position) and a quarternion (rotation). I get the front, right and up vectors of the camera by applying the quaternion to the ...
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5answers
4k views

Why should quaternions exist?

Why do quaternions exist? I want to believe they exist, but all I can think of are reasons they should not exist. These are my reasons. The quaternions are defined by the following equation: ...
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1answer
96 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by John Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic ...
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0answers
47 views

Angular Velocity calculation

I am trying to calculate the time derivative of the quaternion from the following paper: Robotics and Biomimetics (ROBIO) See equation 1 below: ...
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1answer
27 views

Calculate hand position from upper-and lower arm's orientation

I've got two unit quaternions in world space representing the lower- and upper arms orientation. The lengths of upper- and lower arm are known. How can i calculate the hand-position relative to the ...
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0answers
46 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and ...
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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 ...
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1answer
33 views

Rotation in 4D?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is ...
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8answers
5k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
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1answer
44 views

Expressing unit quaternions in three degrees of freedom

Short version of question: I am trying to use quaternions to avoid gimbal-lock, but I don't know how to express unit quaternions using three degrees of freedom without re-introducing Euler angles and ...
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1answer
52 views

quaternions - understanding a formula

Quaternions are new for me. I am trying to understand the following formula: What are: $\large{q^x}$ ? I don't think it is a power. $\large{q^t}$ ? just a transposition of the quaternion $q$? ...
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How to compute angular velocity given a set of unevenly spaced quaternions/direction cosine matrices

I have the time evolution (unevenly spaced) of around 1000 quaternions which provides the transformation from an inertial coordinate system to a body fixed. My goal is to obtain the angular velocity ...
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0answers
52 views

Quaternion rotation intuition

Say the quaternions real and imaginary part are written as $(q_1, \vec q)$. One useful multiplication property is $qr=(q_1r_1 - \langle\vec q, \vec r\rangle, q_1\vec r + r_1\vec q + \vec q \times \vec ...
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4answers
463 views

Matrix Representation of Octonions

Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following $$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ...
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1answer
45 views

Does symplectic K-theory $KSp$ have products?

The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the ...
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0answers
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Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
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2answers
39 views

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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0answers
37 views

What does 'unit-norm quaternions' mean?

What does 'unit-norm quaternions' mean in this context here? Thanks in advance.
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1answer
34 views

Could someone explain the notation of the average of quaternions equation?

The equation has some notation that is difficult to find the meaning for. It is equation (3) in the paper 'Quaternion Averaging' by F. Landis Markley, et al. on page 3 under 'The Average Quaternion'. ...
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1answer
46 views

What does the subspace of SO(3) corresponding to zero yaw look like

Background : I'm solving an engineering problem where I have to estimate the orientation of a body in 3D space. Usually, I use quaternions to do this, but I have to consider a special case where I ...
3
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1answer
5k 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
55 views

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$?

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$? This question stems from Stillwell's Naive Lie Theory where he states that a quaternion $t$ of absolute value ...
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0answers
23 views

Constructing a periodic function around $f(t): R \to R^3$

Definitions: $f(t): R \to R^3$ $\hat{\bigtriangledown}f(t) := \frac{df(t)}{dt} \frac{1}{|\frac{df(t)}{dt}|}$ $\vec{a}e^{\vec{b}x} := \vec{a}\cos(x) + \vec{b}\sin(x)$ $\vec{a}, \vec{b} \in H$ Is ...
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1answer
51 views

Quaternions and rotation

Basically, I am programming an iOS application where I use attitude of the device in quaternion format. Problem is following: Practically: I have a device that does a measurement #1 of magnetic ...
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2answers
63 views

Clarification of definition of “inverse” with quaternions

From what I understand, the inverse of a matrix only exists if the matrix is square. I recently learned however that the inverse of a quaternion is the quaternion vector (1xn dimensions) where each ...
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3answers
42 views

Multiplication of quaternion vectors

Upon watching a lecture on quaternions (Youtube link), I came across the following math: $$(a,\vec{v})(a,- \vec{v})=(a^2+(\vec{v}\cdot \vec{v}),-a\vec{v}+a\vec{v}+(\vec{v}\times \vec{v}))$$ where $a$ ...
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4answers
79 views

Why is it that with quaternions $ij \neq ji$?

I've been using rotations in 3d space lately for simulations. Today I came across the quaternion, which from what I understand will be a much better alternative to my cross/dot product methods. Now I ...
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2answers
922 views

Rotating a 4 dimensional point?

I'm trying to rotate a 4 dimensional point (w,x,y,z). So far I've been rotating around planes (wx,xy,yz,zw,wy, and xy), but the order in which I do these rotations changes the results and can ...
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1answer
463 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 ...
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0answers
28 views

quaternions are less versatile than matrix?

I am doing some research looking should I implement quaternions or matrices. What I've seem to come across is that while quaternions can be better for doing smooth rotations and dual quaternions can ...
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1answer
30 views

Correspondence between rotations and pairs of antipodal unit quaternions

I'm having some trouble understanding how rotations of $\mathbb{R}^3$ correspond to antipodal pairs of unit quaternions. In section 1.5 of his Naive Lie Theory, John Stillwell proves the theorem that ...
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1answer
614 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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0answers
33 views

4x4 Matrix with homogeneous coordinates

I learn for a linear Algebra exam and I have the task: "What is the $4\times 4$ matrix , a rotation about the $\pi/3$ describes in homogeneous coordinates about the axis? What is the ...
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1answer
39 views

why is representing rotations through quaternions more compact and quicker than using matrices??

According to the wikipedia page on Quaternions: The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. However, I have to ...
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1answer
29 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
348 views

Exponential Function of Quaternion - Derivation

The equation for the exponential function of a quaternion $q = a + b i + c j + dk$ is supposed to be $$e^{q} = e^a (\cos(\sqrt{b^2+c^2+d^2})+\frac{(b i + c j + dk)}{\sqrt{b^2+c^2+d^2}} ...
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0answers
17 views

Quaternion - trigonometric form - $q=\cos \theta +u \sin \theta$ Components for $u$?

It is proven that a quaternion has the following trigonometric form: $$q=\cos \theta +u \sin \theta.$$ My question is: Which are the components of the $u$? Thanks!
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0answers
19 views

Quaternion and Euler angles small angle proof

Let's start with a quaternion $q = \begin{bmatrix} q1 & q2 & q3 & q4 \end{bmatrix}^T$. Where $q_4$ is the scalar part, which is equal to: \begin{equation} q_4 = cos(\frac{\alpha}{2}) ...
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1answer
32 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
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1answer
49 views

3D Extension of a Fun Geometric Series Puzzle

After being inspired by this question, and in particular Semiclassical's excellent response and generalisation, I thought of another generalisation to a 3-dimensional plane.: Suppose you start at ...
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1answer
82 views

Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)

I am trying to solve the following exercise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. ...
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1answer
74 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
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1answer
74 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
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1answer
27 views

Dot product of of quaternion-rotated vectors

I'm reading http://people.csail.mit.edu/bkph/articles/Quaternions.pdf and it says "it is easy to show that the operation preserves dot-products." on the page 3. But how is it done? I tried to make a ...
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0answers
15 views

How to transform Tait-Bryan-Angels to different rotation orders?

I am having trouble finding or understanding how to get Tait-Bryan-Angels from a rotationmatrix. I have a given rotation matrix $R_q$ which was calculated from the quaternion $q$. I know how to ...
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1answer
56 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
2
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0answers
392 views

$4D$ rotations and quaternions

I have a question about $4D$ rotation: I programmed a little $4D$ game and I used the classical hyper-sphere coordinates, to rotate a vector. It works, but it has some problems: (just for clarity I ...
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1answer
138 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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0answers
56 views

Calculating two rotation angles from xyz coordinates for dummies

This post is a bit verbose so that others who come later may benefit from my thick headedness. I am attempting to construct a primitives composition and constructed solids geometry parser/processor ...
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1answer
52 views

Multiplication of a quaternion and a scalar to produce a vector?

I am looking at someone else's code, and in it they have a quaternion multiplied with a scalar in order to produce a vector. He used the boost library, and can't find exactly where they defined the ...