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

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1answer
52 views

Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
6
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2answers
194 views

Alternative quaternion multiplication method

Given two quaternions, $a+bi+cj+dk$ and $e+fi+gj+hk$, their product (w.r.t. their given order) would normally be given by ...
2
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1answer
427 views

Converting quaternion or $4\times 4$ matrix to $3\times 3$ matrix representation.

I'm working on some code that manipulates an Axis-Aligned Bounding Box (AABB), so it always encompasses the object it borders. I use a $3\times 3$ matrix to re-size the box when it rotates. The ...
3
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2answers
59 views

What is the right name for the space occupied by a quaternion

I have a little problem wrapping my head around quaternions, in particular I have problems about how to pair the usual "3D algebra" with the theoretic vision of a quaternion. I know that informally a ...
0
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0answers
58 views

Conjugation of quaternions

This proof is extreeeeemely boring, but still I must get it right. Let $x = x_0 + x_1 i + x_2 j + x_3 k \in \mathbb{H}$ (the Hamilton quaternions). Conjugation is defined as: $$x^\ast = x_0 - x_1 i - ...
0
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1answer
77 views

Velocity vector transformations with respect to a global frame of reference

This seems like it should be a simple problem, but I've been stuck on it for about a day now. It's technically a programming problem, but I'm posting it here because the root of the problem is really ...
1
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1answer
115 views

The major differences between dual-quaternions and screw theory

I am working on a project involving the motion of rigid body. From the literatures, I found two main tools, namely the dual-quaternions and screw theory. May I ask what are the major differences ...
0
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1answer
53 views

Align the cube's nearest face to the camera

I have a cube and 4x4 transformation matrix Cube is rotated randomly I need to find the nearest face of cube regarding to camera and rotate the cube by aligning that face to the camera. How can I do ...
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2answers
367 views

Is there an algebraic closure for the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r ...
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1answer
202 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 ...
0
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1answer
94 views

Determine similarity between two sequence of quaternions while allowing a degree of freedom around Z axis

A person holds his phone and rotates it in space in a sequence. I am able to obtain a sequence of quaternions from the phone's motion sensors representing the rotation of the phone from the phone ...
0
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2answers
56 views

Find all Quaternions Satisfying..

Let H be the skew field of quaternions. Find all quaternions x satisfying $(i + j)x(i + k) = 2$ I'm having trouble figuring out what to do with this question. I know the "i j k i j k" formula for ...
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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 ...
7
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1answer
150 views

Can quaternions be useful for integrals?

Lets assume we want to find a closed form for $\int_0^1 f(x) dx$ where $f(x)$ is a real-analytic function. There are many techniques to find that. Some include contour integration on the complex ...
2
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1answer
165 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 $H \cong K \unlhd M$ ...
1
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1answer
116 views

Know if a 4x4 matrix is a composition of rotations and translations (quaternions)

I am using quaternions to describe 3D transformations. A position in space is representated by a (x,y,z,1) vector, and a transformation by a 4x4 matrix, following quaternions logics as far as I could ...
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1answer
106 views

What is the general definition of the conjugate of a multiple-component number?

I know that the conjugate of a binomial is the negation of the second part. So the conjugate of (a + b) would be (a - b). I know that the conjugate of a complex number (a + bi), similarly, is (a - ...
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1answer
146 views

What is the magnitude of a dual number? I'm not finding information on this.

I'm investigating dual quaternions and am having to learn a lot of stuff myself because I'm finding very few resources on the mathematical background. I know that the magnitude of a dual quaternion ...
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2answers
183 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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0answers
65 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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2answers
1k 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}} ...
3
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1answer
107 views

Finding $J$ such that this diagram commutes

DISCLAIMER: This is not homework. I did this exercise here and need someone to check if my work is correct: Is it possible to find a matrix $J\in M_{2n}(\mathbb C)$ such that the following diagram ...
5
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2answers
209 views

One more question about mapping quaternionic matrices into real matrices

Real matrices that lie in the image of the inclusion homomorphism $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ are called complex linear real matrices. It is easy to see that a real matrix is ...
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2answers
55 views

On the inclusion homomorphism for quaternionic matrices into complex matrices

My thoughts / background information: It is easy to find an inclusion homomorphism for complex matrices into real matrices: considering the one dimensional case note that multiplying a complex number ...
0
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1answer
32 views

Quaternion Negative Unity

I'm reading Hamilton's Paper on Quaternions. Found here http://www.emis.de/classics/Hamilton/OnQuat.pdf. On page 5, the first statement of 7, says that there are only two different square roots of ...
0
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1answer
35 views

Question about Hamilton's Quaternion Paper

So I was reading Hamilton's paper on quaternions. http://www.emis.de/classics/Hamilton/OnQuat.pdf. On page 2, I'm having trouble following how QQ' and equations A,B,C lead to equation D. My main ...
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33 views

Basis for quaternionic functions

We know that the set of functions $\{1,\cos x, \sin x, \cos 2x, \sin 2x, ... \; | \,x \in \mathbb{R} \}$ is a basis in the space $L^2_\mathbb{R}[-\pi,\pi]$ . Given a quaternion $z \in \mathbb{H}$ ...
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0answers
43 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 ...
2
votes
2answers
88 views

Why is this algebraic manipulation of quaternions incorrect?

I know that for quaternions, $$i^2=j^2=k^2=ijk=-1$$ I've tried to understand this intuitively as having $i$, $j$ and $k$ represent a rotation about each of three axes. But when I do a bit of ...
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1answer
52 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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0answers
75 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 ...
0
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1answer
29 views

Given unit quaternions $q_0,q_1$, find $q$ such that $q_1 = q^* q_0 q$

I rotate an object in space and find two orientation (unit) quaternions. $q_0 = {}^{M_2}_{M_1} q$ is the orientation at the 2nd position relative to the 1st position, measured in frame M. $q_1 = ...
2
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1answer
146 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 ...
0
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2answers
62 views

How does a quaternion differ from a position in terms of algebraic structure?

For two positions, I can subtract one from another to get a vector; I can take combination of them to get another position. My question is, can I treat quaternions in the same way? To be more ...
0
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1answer
135 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 ...
4
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2answers
424 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 ...
1
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1answer
150 views

Showing $\mathbb{H}$ is isomorphic to a subring of $M_2(\mathbb{C})$ as $\mathbb{R}$-algebras

I'm currently trying to show that the ring $A =\begin{pmatrix} a & b \\ - \bar{b} & \bar{a} \end{pmatrix} $ is isomorphic to the real quaternions $\mathbb{H}$ as $\mathbb{R}$-algebras. ...
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2answers
1k views

Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
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0answers
175 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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1answer
90 views

Quaternions $\Leftrightarrow $ Rotations -Conceptual question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve) $ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y ...
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0answers
60 views

Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...
0
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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 $\vec{v_1}$ ...
2
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1answer
195 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 ...
2
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2answers
97 views

Matrix exponentiation intuition.

What does $x^A$ intuitively mean if $x \in \mathbb{C}$ and $A$ is any matrix? Also, what if we had $x$ being a matrix too? Last but not least, what happens if we have a complex $x$ raised to a ...
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0answers
48 views

Quaternion Integration - And conversion to 3D matrix

I have a rotation matrix let us say $R(t)$ and its quaternion $q(t)$. We know already how to convert a quaternion to rotation matrix. Now if I want find $\int R(t) \ dt \tag1 $ can we do that in ...
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1answer
41 views

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$.

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$. My Attempt: $(\rightarrow):$ $$\alpha i=ai-b-ck+dk \Rightarrow -b+ai+dj-ck$$ ...
0
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1answer
672 views

Quaternion , DCM , Euler Angles and Rotation Matrix Differences and when to use?

Quaternion , DCM[Direction Cosine Matrix] , Euler Angles and Rotation Matrix Differences and when to use ? All of the above components can represent rotation , so when to use each of them , best ...
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1answer
729 views

Reaching a point B in Cartesian coordinate via Euler angles knows its initial point A Euler angles and Cartesian coordinates

I have a point A:- Known it's Cartesian coordinates (X,Y,Z) and its Euler angle Aka body rotation (R,P,Y) respectively Roll (rotation around X axis) , Pitch (rotaion around Y axis) and Yaw (rotation ...
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0answers
186 views

Conversion of a matrix differential equation for rotation to quaternions

Specifications and Data We have a 3D rotation function $R(t)_{3\times 3}$ and function ${K(t)}_{3\times 3}$, a matrix-valued function that gives skew symmetric matrices as outputs. It has the ...
3
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2answers
144 views

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 ...