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

learn more… | top users | synonyms

2
votes
2answers
56 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 ...
0
votes
0answers
28 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. ...
1
vote
1answer
52 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} ...
0
votes
0answers
42 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
vote
1answer
68 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
votes
0answers
39 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 ...
0
votes
1answer
50 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
32 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 ...
0
votes
0answers
44 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
votes
1answer
70 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
vote
0answers
32 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 ...
0
votes
4answers
96 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
62 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
54 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?
8
votes
1answer
297 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 $q(t),q(0)$ ...
19
votes
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?
8
votes
1answer
341 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
54 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
votes
1answer
60 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
1
vote
0answers
22 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
43 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
1
vote
3answers
81 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
94 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} & ...
1
vote
0answers
8 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
votes
0answers
42 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. ...
1
vote
2answers
61 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 ...
0
votes
0answers
19 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 ...
2
votes
2answers
146 views

Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
2
votes
1answer
62 views

Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $ \mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
0
votes
1answer
77 views

Converting quaternions to spherical angles

Consider a situation where a beam is shot at a cube C from an arbitrary position P. The cube detects the angle of incidence relative to its $ x $ axis. The cube can be rotated and moved, and the ...
1
vote
0answers
50 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 ...
2
votes
1answer
40 views

Need help with this exercise about real division algebra

I am trying to solve the following exrcise: 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. I ...
3
votes
4answers
158 views

Need help determining the pairs of quaternions that anticommute

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 $p,q \in \mathbb H$ such that $pq = -qp$. I ...
1
vote
0answers
39 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 ...
3
votes
1answer
77 views

Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...
1
vote
1answer
139 views

Isomorphism of quaternions with a matrix ring over real numbers

Let $\mathcal A$ be the algebra over the real numbers consisting of matrices of the form $$\begin{pmatrix} z&w\\ - \bar{w}& \bar{z} \end{pmatrix} \ (z, w \in \mathbb C). $$ $\mathcal A$ is in ...
0
votes
0answers
22 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
1answer
28 views

Error performing multiplication of Quaternions

Alright I'm going to try one last time to explain my problem with quaternions and multiplication of two quaternions in specific. This time hopefully I'll get an explanation that makes sense. (I posted ...
0
votes
0answers
26 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
2answers
44 views

Smooth transition between two quaternions?

I am describing the orientation of an object with quaternion $q$. Now I want to describe (animate) smooth transition between orientations of $q_1$ and $q_2$. I was thinking that quaternion $q = q_1 ...
2
votes
1answer
63 views

Is there a (hypercomplex) number system, in which addition is **not** commutative

Now, we all know and love the quaternions, in which multiplication is not commutative, and the fact that, in all Clifford algebras, exponentiation is not commutative. Having looked at the properties ...
0
votes
0answers
28 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 ...
3
votes
1answer
67 views

What is the meaning of quaternion interpolation?

Suppose I take the average between two quaternions, how does one see the meaning of the resulting rotation to make sure it is sensible, unlike interpolating Euler angles? I'm looking for an argument ...
0
votes
2answers
35 views

Rotation between two vectors as a function of time (one parameter rational motion design)

Given a time varying vector: $\mathbf{w}(t) = \mathbf{u} + t\mathbf{v}$ I would like to find a rotation matrix $\mathbf{R}(t)$ that rotates the positive x-axis $[1, 0, 0]^T$ onto the vector ...
5
votes
0answers
172 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 ...
3
votes
0answers
169 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 ...
4
votes
2answers
279 views

What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + ...
0
votes
0answers
62 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 ...
3
votes
1answer
48 views

Proof that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$

In the wikipedia atrticle (http://en.wikipedia.org/wiki/Octonion) it is stated that "one can show that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, ...
6
votes
4answers
149 views

What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?

It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ ...