Tagged Questions

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

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1answer
16 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 = ...
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0answers
37 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 ...
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1answer
27 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 ...
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1answer
47 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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2answers
55 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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0answers
33 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
31 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
24 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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0answers
23 views

Can Euler's formula be extended to Octonions and higher Cayley-Dickson algebras that are non-associative?

Although question is pretty self explanatory, I know that $e^{i\theta}=\cos\theta+i\sin\theta$ and $i$ can be replaced by a unit pure quaternion (which is a root of $-1$ like $i$). Can we do the same ...
0
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1answer
27 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
37 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 ...
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0answers
13 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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0answers
22 views

Quaternion - An equivalent form

Given Data in the problem I have rotation matrices represented by a quaternion $q(t)$ and we are aware of axis of rotation at each point as $\psi(t)$ and angle of rotation $\theta(t)$. I have a ...
2
votes
1answer
35 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
63 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
17 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
33 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
25 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 ...
1
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1answer
64 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
15 views

Convertion to Quaternion

Specifications and Data We have a 3D rotation function $R(t)_{3\times 3}$ and function ${K(t)}_{3\times 3}$ a matrix function that gives skew symmetric matrix as out puts. It means it holds the ...
3
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2answers
85 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 ...
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0answers
36 views

Quaternions- Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represnts the position vector as result of rotation with an angular veclocity $\omega(t)$ in quaternion , then you can make the relationship ...
0
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1answer
28 views

Multiplication of Rotation Matrices in quaternion

Given Data and specifications NB : * means multiplication Suppose we need to rotate a point $P = \begin{pmatrix} x\\ y\\ z \end{pmatrix}$ with rotation matrix ${Q}_{3\times3}$ then what we do is ...
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2answers
47 views

Unit Quaternion to a Scalar Power

I'm trying to modify a physics engine for efficiency. Currently, as objects move around the world, their orientation (a quaternion) is updated every frame, by multiplying by the rotation (another ...
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0answers
17 views

Translating Quaternion rotation from one frame of reference to another.

I have been having issues getting around this for quite a few days. I will appreciate any input or advice. I have a sphere (A) with an applied axis rotation of lets say -45 degrees around the Z-axis. ...
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1answer
49 views

Exercise 5.8 from Lie Group, Daniel Bump

In the exercise 5.8 Bump has asked to prove that the group $Sp(4)$ over complex numbers, which is usual complex embedding $U(4)\cap Sp(4,\mathbb{C})$, can be described by, $$\left\{\begin{pmatrix} ...
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0answers
29 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 ...
1
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1answer
55 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
0
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0answers
34 views

Quaternion Integration - Initial value problem

We have a standard form of quaternion integration equation $$ q(t) = q(t_0) \exp\left(\frac 12 \int_{t_0}^t \mathbf{\omega}(\tau) d\tau\right),\tag 1 $$ For reference you can check equation (42) in ...
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0answers
31 views

Quaternion Equivalence

Assume $R_{3\times3}$ is a rotation matrix. Question Is it true that there exists two quaternions representing this same rotation matrix $R_{3\times3}$ ? Hint : $\theta = \arccos\left( ...
0
votes
1answer
28 views

Quaternionic representation

Let $V$ be $G$-representation over quaternions $\mathbb{H}$. How to show that $$ \mathbb{H} \otimes_\mathbb{C} V $$ is canonically isomorphic to $V \oplus V$ as representation over $\mathbb{H}$? In ...
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0answers
34 views

Quaternion Solution of the Rotation Equation

I am trying to make a connection between a 3-d vector ODE with a quaternion ODE and a possible solution in quaternion. In the following, a vector $v$ in $R^3$ is interpreted as the vector part of the ...
3
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0answers
58 views

Showing that $\mathbb{H}^{*}$ maps onto $\mathrm{Aut}(\mathbb{H})$

To show that $\mathbb{H}$ maps onto $\mathrm{Aut}(\mathbb{H})$, where $\mathbb{H}$ is the Hamilton quaternions, I thought it'd be pertinent to show first that the subgroup of inner automorphisms of ...
1
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0answers
30 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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4answers
80 views

Rotating Frame and Angular Velocity

We have an equation $ \frac{dr}{dt}=\Omega \times \bf r \tag 1$ SPECIFICATIONS $\times$ means cross product,$\Omega$ constant angular velocity,${\bf r}$ is the postion vector of an object Given ...
2
votes
1answer
53 views

Do all automorphisms of $\mathbb{H}$ preserve the norm of an element?

Do all automorphisms of $\mathbb{H}$-- the Hamilton quaternions-- preserve the norm of an element? I can't seem to answer this question without using the not-so-elementary fact that all automorphisms ...
0
votes
2answers
52 views

Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
7
votes
1answer
184 views

Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
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6answers
2k views

What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
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1answer
331 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
3
votes
1answer
53 views

Finding quaternion that transforms to particular basis

I want to find a quaternion $x \in{\mathbb{H}} $ that transforms (rotates) the $ i,j,k $ basis to a particular basis. In equations: $$ x i x^{-1} = a_1 $$ $$ x j x^{-1} = a_2 $$ $$ x k x^{-1} = a_3 ...
0
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1answer
47 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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0answers
17 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 ...
0
votes
1answer
42 views

There is no group whose quotient by the center is isomorphic to the quaternion group [duplicate]

I have to prove that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. Anytone can give me an idea to begin? thanks
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vote
3answers
80 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
3
votes
3answers
90 views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
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0answers
7 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 ...
0
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0answers
37 views

Rotation plane on the sphere (quarternion)

I asked similar question on stackoverflow but still no answers.http://stackoverflow.com/questions/25185329/image-rotation-with-the-gyro-data-math I assume it is more math than programming problem. ...
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vote
2answers
56 views

Derive a quaternion from three axis

My problem originates from some code that I'm writing to parse an obscure file-type in which a geometric entity is defined in it's own 'local space', and a rotation and translation are provided to ...
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0answers
17 views

Multilinear quaternion interpolation

I'm looking for literature to study more on multilinear quaternion interpolation. Looking for 'trilinear interpolation' and 'tricubic interpolation' on Google Scholar or arxiv doesn't yield much ...