2
votes
2answers
62 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 ...
0
votes
0answers
15 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 ...
0
votes
0answers
14 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 ...
0
votes
0answers
32 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
votes
1answer
27 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 ...
1
vote
1answer
52 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
33 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
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( ...
1
vote
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 ...
0
votes
4answers
78 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 ...
0
votes
2answers
49 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?
3
votes
3answers
86 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} & ...
0
votes
0answers
34 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. ...
2
votes
2answers
87 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. ...
3
votes
0answers
104 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 ...
1
vote
1answer
131 views

Quaternion and Matrix

I have a quaternion for rotation and a matrix for changing axis(change coordinate from camera to my rendering scene ). I have tested two method and i except to have equal resuls but results are ...
0
votes
1answer
75 views

Quaternion isn't normal!

I use the following scheme: Quaternion: [ w, x, y, z ] Then I use Euler Angles and convert it to a quaternion, as following: ...
1
vote
0answers
96 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
2answers
52 views

Show by Example that $\mathbb{H}^n$ to $\mathbb{H}^n$ is not necessarily $\mathbb{H}$-linear

Show by example that for$ A \in M_n \mathbb{H}, L_A : \mathbb{H}^n \rightarrow\mathbb{H}^n $ is not necessarily $\mathbb{H}$-linear So I thought it would be linear by definition. Because if we have $ ...
0
votes
0answers
19 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
1answer
205 views

3D Positioning Vectors to Matrix & Quaternions

This is also programming related, but I think it's more about math than anything else. I have an Object which I want to represent it's 3d position, rotation and scale with vectors, and then I need to ...
0
votes
0answers
93 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 ...
3
votes
1answer
336 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
2
votes
0answers
34 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 ...
0
votes
1answer
84 views

Quaternion Matrix Multiply

My question is about applying rules of quaternions to quaternion matrices. I know that for some rotation quaternion q = [w, x, y, z], I can find the rotation of ...
3
votes
3answers
218 views

Are there different conventions for representing rotations as quaternions?

I am trying to understand how quaternions are represented as rotations, in particular how to convert from a quaternion representation to a rotation matrix. The following paper by Diebel gives an ...
5
votes
2answers
535 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
0
votes
1answer
35 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
4
votes
3answers
330 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
3
votes
1answer
203 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
3
votes
2answers
59 views

What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ...
1
vote
1answer
46 views

How does one derive this rotation quaternion formula?

given an angle and an axis, the corresponding quaternion can be computed like this. $w = \cos( Angle/2)$ $x = \text{axis}.x * \sin( Angle/2 )$ $y = \text{axis}.y * \sin( Angle/2 )$ $z = ...
1
vote
0answers
168 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, ...
2
votes
1answer
278 views

Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
16
votes
8answers
3k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
0
votes
1answer
132 views

strange matrix -> quaternion conversion problem

I find two different rotation matrices are mapped to a single quaternion. $$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ 0 & ...
2
votes
3answers
344 views

Matrix Representation of Octonions

Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following $$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ...
5
votes
1answer
796 views

Using quaternions instead of 4x4 matrices for transformations

I'm interested in implementing a clean solution providing an alternative to 4x4 matrices for 3D transformation. Quaternions provide the equivalent of rotation, but no translation. Therefore, in ...
1
vote
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 ...
1
vote
2answers
2k views

How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?

How would I create a rotation matrix that rotates X by a, Y by b, and Z by c? I need to formulas, unless you're using the ardor3d api's functions/methods. Matrix is set up like this ...
5
votes
3answers
870 views

On multiplying quaternion matrices

Both matrix multiplication and quaternion multiplication are non-commutative; hence the use of terms like "premultiplication" and "postmultiplication". After encountering the concept of "quaternion ...