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

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1answer
36 views

How to decompose a unit quaternion into 3 Tait-Bryan quaternions instead of 3 real numbers?

I'm familiar with the formulas for decomposing a unit quaternion $Q$ into chained Tait-Bryan angles $\phi\theta\psi$ (Wikipedia has the formulas for the $zyx$ chain here), but I'm looking to instead ...
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1answer
64 views

Why does rotation by a quaternion require multiplying two times?

Given a vector $p$, to rotate it by a quaternion $q$, we use the formula: $$p' = q p \hat{q}$$ where $\hat{q}$ is the conjugate of $q$. But if we use rotational matrices, then it's just $$p' = Rp$...
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1answer
45 views

Calculating a quaternion that represents a given rotation

This is the first time I'm attempting to do a quaternion and I am not quite getting the concept. This is part of a 3 calculation homework question The initial question is Given a 3-D point at ...
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2answers
73 views

Why do division algebras always have a number of dimensions which is a power of $2$?

Why do number systems always have a number of dimensions which is a power of $2$? Real numbers: $2^0 = 1$ dimension. Complex numbers: $2^1 = 2$ dimensions. Quaternions: $2^2 = 4$ dimensions. ...
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0answers
17 views

Degree of quaternion product composed with two maps of $S^3$

Let $S^3$ denote the unit quaternions with multiplication $\mu:S^3\times S^3\rightarrow S^3$.Show that if $f_1,f_2:S^3\rightarrow S^3$ are given maps,that the composition $$S^3\xrightarrow{f_1\times ...
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0answers
15 views

Relative positioning using quaternions

Say I have quaternion $q_1$, which I have achieved from my IMU module. I want to state that current position is $initial$. Then I want to compute Euler angles relative to this initial position at the ...
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0answers
35 views

What algebra do you get if you switch the sign of one pair of anticommuting quaternion products?

What are the properties of an altered quaternion algebra defined by: ii = jj = kk = -1, ij = -ji = -k, ik = -ki = +j, jk = -kj = +i, Is it associated with any manifold?
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37 views

Extract the angle of rotation from a unit quaternion

Sorry for boring you my friends before the spring vacation. I am haunted by a simple problem of how to extract rotation angle from a unit quaternion. Suppose $a$ is a unit quaternion which takes the ...
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2answers
72 views

Quaternion for beginner

QUATERNION ROTATIONI have 2 points in 3d space ,point A= <5,3,6> and B=<8,2,3> I want to rotate "point B" by 30 degree from point A. how do I solve this question using quaternion. Plz, ...
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20 views

Quaternions and Rotations [duplicate]

I have 2 points in 3d space ,point A= <5,3,6> and B=<8,2,3> I want to rotate "point B" by 30 degree from point A. how do I solve this question using quaternion. Plz, explain the steps....
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0answers
39 views

Calculate ψ knowing object orientation in 3D through forward and up vector

I've got a so called right, up, forward tridimensional reference plane and an object $P$ in it. Its orientation in space is defined by two vectors, forward and up: -forward gives azimuth $θ$ and ...
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0answers
25 views

Translation offset when converting Matrix to Dual Quaternion skinning

i have a problem with dual quaternion skinning. if i convert my matrixes to dual quaternions i have a fixed offset from the bones (rotation is correct). if i transform everything with identity, than ...
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1answer
41 views

Every Hamiltonian group contains a subgroup isomorphic to $Q_8$

I read somewhere that every Hamiltonian group (a non abelian group with every subgroup normal) contains a subgroup isomorphic to quaternion group. But I cannot find its proof anywhere on net or in ...
2
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1answer
133 views

Normalizing a quaternion

How do I normalize a quaternion $$q=w + \mathbf ix + \mathbf jy + \mathbf kz = a + v$$ ? I already know: The normalized quaternion is called unit quaternion and can be calculated in this way: $$U_q = ...
3
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1answer
62 views

Quaternion exponential problem

I have problem with Euler´s form of quaternion. My quaternion $q=\frac{1}{\sqrt{2}}i+\frac{1}{\sqrt{2}}j,$ so $q^2=-1$, because $$q^2=(\frac{1}{\sqrt{2}}i+\frac{1}{\sqrt{2}}j)(\frac{1}{\sqrt{2}}i+\...
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0answers
51 views

Differences between Quaternion integration methods

I've implemented a Quaternion Kalman filter and i have the choice between multiple way to integrate angular velocities. The goal is to predict futur orientation $q^{n+1}$ from current orientation $q^{...
2
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1answer
53 views

Quaternion from global space to local space

I've searched but have not found a response for this question specifically. I have a smartphone with a sensor that gives me a quaternion representing its absolute rotation relatively to the following ...
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1answer
44 views

Why do two different quaternions appear to have the same rotation?

When using a Quaternions I've noticed something I don't quite understand. If I'm rotation $\frac{\pi}{2}$ radians on the Y axis it goes from $[0,0,0,1]$ to $[0,\sqrt{2},0,\sqrt{2}]$. A rotation of $\...
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1answer
46 views

difference between 2 quaternions

I'm trying to calculate quaternions relative to a given orientation. It is easiest for me to explain my intentions by means of an example: Suppose you have a vector $v1=[0,0,1]$ and I want to rotate ...
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1answer
35 views

Proof of quaternion algebra being simple using norm

I was wondering if the following simple (pun unintended) proof of the quaternion algebra $A=\left(\frac{a,b}{F}\right)$ being simple is valid. I saw many more complicated proofs online, eg: Proof ...
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23 views

Check that $u_4\bar{u_3}u_2\bar{u_1}=i$ and $\bar{u_1}u_2\bar{u_3}u_4=1$ so the product of the four reflections is indeed $q \to iq$

This is an exercise from "Naive Lie Theory" and $u_1, u_2, ...,u_4$ are the unit quaternions. I have read the section many times but still don't understand. Can someone explain the material and solve ...
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1answer
47 views

Understanding rotations of $\mathbb{R}^4$ and pairs of quaternions, showing a rotation is a product of reflections in hyperplanes

I am working through Stillwell's "Naive Lie Theory" and am completely stumped by the questions in this section. An example of one of the questions is Show that the rotation that sends $1$ to $i$, $i$...
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1answer
58 views

Book(s) on Algebras (Quaternions)?

Well, lately I've been looking for a book on quaternions but I've realized that quaternions are a particular case of the named Algebras(I think Geometric Algebra). Since here, I've found all kind of ...
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3answers
80 views

Elements of order 2 in the special linear group

I am trying to show that the unique Sylow $2$-subgroup of the special linear group $SL(2,\mathbb{F_{3}})$ is isomorphic to the quaternion group $Q_{8}$. Call the unique Sylow $2$-subgroup $P$, and ...
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0answers
54 views

Quaternion Kalman Filter Process Noise

I'm implementing a extended Kalman filter using quaternions. I've extended this paper to deal with my custom observations. My state space is analogous to the one in the previous paper : $ \mathbf{...
0
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1answer
29 views

Problem with converting rotation representations (quaternion, axis-angle, etc)

I have a computer device - a 3D pointer (Sensable Phantom Omni). It returns cartesian position (X,Y,Z) and orientation quaternion (x,y,z,w). Now I have a 3D visualization software (PyMOL) and I need ...
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1answer
24 views

Quaternion for transforming one frame to other?

I am new to quaternions and learning how they can replace rotation matrices. I know that we can use rotation matrices to describe a transformation from one frame to other. Where one may be a rotated ...
0
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1answer
105 views

Convert quaternions to xyz degrees

I knew quaternion for the first time a few days ago and I still don't get the way it works even when reading explanations. All I want to do is to make a subtraction between two quaternions and convert ...
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0answers
22 views

How do I calculate Jacobian of formula containing quaternions and vectors?

I am facing a problem in robotics where a robot is localized in 3D-space to build up a map simultaneously (see SLAM, e.g. [1]). One approach is to build up a graph of poses $x_i$ and transforms $z_{ij}...
0
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1answer
53 views

Rotation addition with quaternions

My task is: "Describe rotation $S \circ R$ by axis and angle, where $R$ is rotation around $(0,1,1)$ by 90 degrees, and $S$ is rotation around $(1,-1,0)$ by 90 degrees." I should use quaternion ...
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1answer
652 views

Quaternion - Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). And I am trying to calculate the angle of rotation around all the three axes. I have tried may methods but not getting the ...
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2answers
49 views

Extracting the Axis a Quaternion is rotating around from the Quaternion itself Directly

Quaternion has components X, Y, Z, and W. If you created a Quaternion with input being a 3D Vector representing the axis (X,Y,Z) and a floating point number representing the amount to rotate around ...
2
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2answers
45 views

multiplication of quaternions is like complex numbers multiplication?

Suppose $p = z + j w $ where $z = x_0 + i x_1$ and $w = x_2+ix^3$. Let $q = \alpha + j \beta $ where $\alpha = y_1 + i y_2$ and $\beta = y_2 + i y_3$. How can we multiply $p$ and $q$. Is is just like ...
2
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1answer
28 views

calculating the orientation of an object

If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to know/...
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2answers
104 views

What is the $\sqrt{-1}$ when working in a quaternion space?

I dont think I really need to elaborate, do I? If you know what quaternions are then you know there are several imaginary-value options to choose from, or axes, along which the $\sqrt{-1}$ may exist. ...
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1answer
28 views

Rotation matrix between two similar cuboids using their upper sides ( and the planes defined by these sides)

I have two different images and with them an estimation of two planes ( defined in the same system). I would like to get the rotation matrix, quaternion or euler angles of a surface within this planes....
0
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1answer
25 views

How to rotate in quaternions but for 2d version for arbitrary angle?

I am trying to understand the idea behind rotating in quaternions, but first I want to understand the math for 2d rotation. I saw some youtube videos, and I know that for 2D, a point in 2D can be ...
5
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1answer
96 views

Quaternions: Why is the angle $\frac{\theta}{2}$? [duplicate]

The equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta 2\right)...
2
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1answer
29 views

Why is the set of units of integer quaternions isomorphic to the quaternion group of order 8?

Let's say that I've got a ring $V$ of integer quaternions of the form $\mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$. Now assume that there exists an element $a = a_1 + a_2i + a_3j + a_4k$ ...
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1answer
54 views

Why does $ab=ba=1$ imply ${a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 = 1$?

Let's say that I've got a group $V$ of integer quaternions of the form $\mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$. Now assume that there exists an element $a = a_1 + a_2i + a_3j + a_4k$ ...
3
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0answers
52 views

Davenport's Q-method (Finding an orientation matching a set of point samples)

I have an initial set of 3D positions that form a shape. After letting them move independently, my goal is to find the best rotation of the original configuration to try to match the current state. ...
2
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1answer
40 views

rotation to quaternion matrix handeness

I've understand that quaternions do not have handness but rotation matricies derived from unit quaternions does. The following formula is given by wikipedia for quaternion to rotation matrix ...
0
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1answer
47 views

Does the commutative property not apply when multiplying Quaternions

I've been looking up Quaternion multiplication and many resources have stated that $j*k=i$ and also in other sources I've found $k*j=-i$ But I have not found any sources stating $j*k=-i$ and I ...
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1answer
34 views

How to prove associativity of quaternion multiplication using scalar and vector form?

In scalar and vector form, a quaternion can be represented as $a=(q_0,{\bf{q}})$. The definition of quaternion multiplication is: $ab=(q_0,{\bf{q}})(p_0,{\bf{p}})=(p_0q_0-{\bf{q}}\cdot{\bf{p}},q_0{\bf{...
0
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1answer
22 views

Extended Kalman Filter - Determining the state vector

I am trying to make sense of a paper that defines a EKF approach to estimating the heading of a device. The paper says the following: The state vector $x$ for the filter is composed of the rotation ...
0
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0answers
73 views

4D representation of finite group and projection operator

Suppose I have a finite group, say $C_2$, and its Cartesian representation $ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \...
10
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3answers
780 views

Are j and k on different imaginary planes than i?

I'm trying to understand Quaternions. So I understand that a Quaternion is written like $xi+yj+zk+w$. I also understand that $i^2 = j^2 = k^2 = ijk = -1$, and how that can be used to derive equations ...
2
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0answers
48 views

Is there a way to prove vector triple product from quaternion multiplication?

For pure imaginary quaternions $u, v, w$, is there a way to prove the vector triple product $u\times(v\times w) = v(u\cdot w) - w(u\cdot v)$ from the relation: $$uv = -u\cdot v + u\times v \text{ for $...
2
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1answer
72 views

Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
0
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0answers
90 views

Angular velocity computation

Say I have two different unit quaternion $Q1$ and $Q2$ representing two different orientations in 3D space. How can I compute the angular velocity $\omega$ that would produce a rotation from $Q1$ to $...