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

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8
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433 views

Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
5
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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 ...
4
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0answers
58 views

Hamilton's letter to his son

I'm looking for a better reference on this letter from Hamilton to his son where he wrote about his discovering on Quaternions. I'd like to read, if it is possible, a scanned version of the letter. ...
3
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0answers
15 views

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 ...
3
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0answers
363 views

Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b ...
3
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0answers
162 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
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0answers
115 views

The geometry of $\operatorname{PSO}(4)$ and the quaternions

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
2
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51 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 ...
2
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0answers
101 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
2
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0answers
41 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
2
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50 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
2
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36 views

Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order ...
2
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0answers
161 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 ...
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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0answers
31 views

Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors. What I would like to know is angles ...
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0answers
33 views

Grothendieck group of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such ...
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0answers
51 views

How can I transform coordinate systems based on quaternion data?

I have a single rigid body object, and its orientations in quaternion with respect to two coordinate systems, each is called original and prime, respectively; therefore, I have two quaternions ...
1
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0answers
27 views

Division algebra over 2-adic fields

Let $D$ be the quaternion division algebra and $O$ be a maximal $\mathbb{Z}$-order in $D$, say the Hurwitz quaternion integers. It can be proved that $D$ and $O$ split at odd primes, that is ...
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51 views

Clifford algebra and Spin group of 4-dimensional Euclidean space

I’m seeking for a straightforward construction of well-known $\mathrm{Spin}(4) = \mathrm{Spin}(3)\times\mathrm{Spin}(3)$ isomorphism using geometric algebra-based definition of “Spin”, without ...
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0answers
83 views

Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
1
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0answers
52 views

Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
1
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0answers
50 views

Inverse of a $2\times2$ matrix with noncommuting entries

I want to invert a $2 \times 2$ matrix with quaternionic entries. Since non-commutativity, the determinant is not well defined and I've seen in http://en.wikipedia.org/wiki/Quaternionic_matrix that, ...
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0answers
41 views

A corollary of Niven

Please proof corollary of Niven: For $a \in D\backslash R$, the equation ${t^n} = a$ has exactly $n$ solutions in $D$, all of which lie in $R\left( a \right)$, in there $R$ is a real-closed field and ...
1
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0answers
60 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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0answers
101 views

3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
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205 views

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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0answers
109 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
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0answers
36 views

Cross product uniqueness

I have following relationship between vectors $A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$ Given Data " ' " means derivative ...
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45 views

Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this ...
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0answers
15 views

Find path between two attitudes subject to body rate constraints

Here's my problem. I have an initial orientation and angular velocity of a body and a final orientation and velocity occurring at a specified time in the future. I have control over how input ...
1
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0answers
63 views

can quaternions be expressed in terms of tensor products?

QUESTIONS does this arithmetic check out? if so, is there a geometric interpretation? note: my aim was to try to find a very simple but non-trivial example which might help me begin to understand ...
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0answers
41 views

Identifying $\mathbb H^n$ with $\mathbb C^{2n}$

Let $X \in M_n(\mathbb H)$ (Hermetian field). It is possible to make $\mathbb H^n$ into a $2n$-dimensional vector space over $\mathbb C$, for example, by embedding $\mathbb C$ into $\mathbb H$ as ...
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0answers
98 views

Prove that $Q$ is a group under quaternion multiplication

Consider the subset $Q$ of the quaternions defined by $$Q=\{1,-1,i,-i,j,-j,k,-k\}.$$ Show that $Q$ is a group under quaternion multiplication. I know to prove something's a group, you must show ...
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0answers
348 views

Transform a vector to global frame and ignore rotation about one axis or Full tilt compensated magnetometer

Good day everyone. I would like to lock the rotation about one specified axis. For example, let`s imagine that we have a quaternion which desribes the orientation of our rigid body relative to the ...
1
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0answers
71 views

Integral elements with predescribed properties in quaternion orders

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
1
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0answers
53 views

Plücker coordinates of the Clifford parallels

Let $$q=\cos\theta+(x_q\textbf{i}+y_q\textbf{j}+z_q\textbf{k})\sin\theta$$ be a unit quaternion parameterised by $\theta\in\mathbb{R}$, where $(x_q,y_q,z_q)$ is fixed and $x_q^2+y_q^2+z_q^2=1$, and ...
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0answers
132 views

Indefinite quaternion algebra over Q

Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Q: If we choose another isomorphism ...
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0answers
99 views

Dual quaternion derivation

I'd like to derivate a dual quaternion \begin{align} \hat{q}&=(1 + \frac{1}{2}\epsilon\vec{t})q \end{align} where \begin{align} q &= e^\vec{w} , \\\vec{w}&=(0, w_1,w_2,w_3)^t ...
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0answers
52 views

How stable is my quaternion interpolation?

after some experimentation to optimize slerp I found that finding the middle between quaternion is rather cheap (for $t=0.5$) in particular: (with $\theta$ the angle between $q_1$ and $q_1$) ...
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0answers
62 views

Quaternion Hermitian diagonalization

How do I go about diagonalizing such a matrix. For example let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11$ and take the matrix: $A = \begin{pmatrix} 11 & -j-3k \\ ...
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0answers
98 views

Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
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0answers
47 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 ...
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0answers
54 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 ...
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0answers
232 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, ...
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0answers
226 views

Quaternions in olympiad 3d geometry

It's known that we can use complex numbers to solve some 2d problems easier than synthetic methods. But, what do you think about using complex numbers in 3d geometry? I've found extend of complex ...
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0answers
486 views

How to convert Yaw, Pitch, Roll and Acceleration value to cartesian system?

I am having readings of Yaw, pitch, Roll, Rotation matrix, Quaternion and Acceleration. These reading are taken with frequency of 20 (per second). They are collected from the mobile device which is ...
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0answers
78 views

Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$. We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and ...
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0answers
182 views

Handedness Convention for Pauli Matrices/Quaternions as rotations in 3-space

Its a fairly standard result that one can represent rotations in 3-space with unit quaternions (or pauli matrices which can be mapped to the unit-quaternions). Working through a related quantum ...
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0answers
3k views

Compute Altitude and Azimuth using either Quaternions or Rotation Matrix or Roll, Pitch and Yaw component

I am struck with a mathematical problem. I want to convert the iPhone device's attitude information which is available in one of the following forms: Quaternion Rotation Matrix Roll, Pitch and Yaw ...
0
votes
0answers
9 views

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 ...