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

learn more… | top users | synonyms

1
vote
1answer
41 views

How can I find a unit velocity vector between two quaternions?

I have two quaternions, $Q_0$ and $Q_1$. I want to find the unit angular velocity vector $w$ that rotates $Q_0$ in the direction of $Q_1$ (shortest path). How can I do this? The analog of what I ...
6
votes
0answers
387 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
votes
0answers
112 views

Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
4
votes
0answers
53 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
votes
0answers
138 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
3
votes
0answers
106 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
111 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
votes
0answers
21 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
votes
0answers
35 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
votes
0answers
36 views

Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
2
votes
0answers
58 views

The angle of an average rotation is $126.5^\circ$?

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
2
votes
0answers
32 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
votes
0answers
112 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
votes
0answers
305 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 ...
1
vote
0answers
31 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 ...
1
vote
0answers
31 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 ...
1
vote
0answers
37 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 ...
1
vote
0answers
108 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
vote
0answers
49 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
vote
0answers
32 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 ...
1
vote
0answers
88 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 ...
1
vote
0answers
44 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 ...
1
vote
0answers
26 views

Enumeration of Hurwitz quaternions of norm p

In "on Quaternions and Octonions" by Conway and Smith, they quote a result by which for each prime norm $p$ there are exactly $p+1$ Hurwitz quaternions of norm $p$. I haven't found any proof of that. ...
1
vote
0answers
44 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 $q_1$) ...
1
vote
0answers
45 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 \\ ...
1
vote
0answers
77 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 ...
1
vote
0answers
37 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 ...
1
vote
0answers
47 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 ...
1
vote
0answers
146 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, ...
1
vote
0answers
198 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 ...
1
vote
0answers
336 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 ...
1
vote
0answers
74 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 ...
1
vote
0answers
162 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 ...
0
votes
0answers
18 views

Set of quaternions that anti-commute

I tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $x,y \in \mathbb H$ such that $xy = -yx$. ...
0
votes
0answers
16 views

Finding component Velocity relative to Velocity

I am trying to find the velocity of an object in a particular Direction based on the current rotation and velocity of that object. I will try and illustrate the example: I have a space craft ...
0
votes
0answers
16 views

Transforming euler rotations to other coordinate system

I have a global $3D$ coordinate system, and it's transform matrix is an identity matrix. I also have an object with it's own, local coordinate system and it's transform is not identity matrix. Now I ...
0
votes
0answers
23 views

Quaternions $\leftrightarrow$ Matrix - Source Code

Can anybody help me with a easy source code for the transformation quaternions $\leftrightarrow$ matrix? Any language programming is ok, doesn't matter if it about C/C++ or java, or written in Matlab. ...
0
votes
0answers
22 views

Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
0
votes
0answers
37 views

How to obtain relative rotation?

I have two rotations, each of which can be described as a roll, pitch, and yaw (in radians): $$ r_1 = (3.14159, 1.57080, 1.6) $$ $$ r_2 = (3.14159, 1.57080, 1.4) $$ I am interested in the relative ...
0
votes
0answers
47 views

Quarternions from MPU and circumference of circles

First I should mention that my math skills are super basic. I do not understand formulas but I do understand pseudo code, C, C++, and other programming languages. I've been working on a electronics ...
0
votes
0answers
31 views

Normalization of Euler angle data

I have head motion data for several speakers. Because not every speaker sat in the exact same position during recording I have to normalize the data. One option to do this, I think, would be to ...
0
votes
0answers
85 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
votes
0answers
17 views

Polar form of Bi-Quatenrion

I have a complex Quaternion(Bi-quaternion) and i want to convert that to Polar form (Euler). Let say we have Fourier transform of a ( bi-quaternion ) like, then how can we get a polar form ...
0
votes
0answers
64 views

Conjugation classes and disjoint union

i have a question about group theory. Given the group $SU_2(\Bbb{C})$, then we have the (spin) homomorphism form $SU_2(\Bbb{C})$ to $SO_3(\Bbb{R})$, call it $\pi$, given by $\pi(q)x=qxq^{-1}$. I can ...
0
votes
0answers
35 views

Solving intersection of 4 quadratic equation with constraints

I want to find the intersection of 4 quadratic equation with the constraint that $\left[\begin{matrix}q_0 \\ q_1 \\ q_2 \\ q_3 \end{matrix} \right]$,$\left[\begin{matrix}p_0 \\ p_1 \\ p_2 \\ p_3 ...
0
votes
0answers
125 views

Optimization of quaternion / Rotation of form Q*P*Qbar=R

I wish to find out a unit quaternion $Q$ (with complement $\bar Q$, s.t. $Q\bigotimes \bar Q = \bar Q\bigotimes Q$=[1 0 0 0]) that will satisfy the following equation $Q\bigotimes P \bigotimes \bar Q ...
0
votes
0answers
36 views

Slicing Up Uniform Random Rotation Quaternions

I'm generating uniform random rotations using quaternions. I am using the method attributed to Shoemake, which is discussed in another post (Uniform Random Quaternion In a restricted angle range): ...
0
votes
0answers
87 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
0
votes
0answers
142 views

how to fine the quaternion between two vectors lies on two different Cartesian coordinate systems

I have two vectors lies on different cartesian coordinate systems. I want to find the quaternion between these two vectors. how I can do that?
0
votes
0answers
2k 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 ...