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

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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 ...
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1answer
30 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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53 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
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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 ...
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2answers
37 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 ...
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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 ...
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2answers
60 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
19 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 ...
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1answer
15 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 ...
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1answer
63 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 ...
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22 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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53 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 $a = a_1 + a_2i + a_3j + a_4k$ ...
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0answers
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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. ...
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1answer
29 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 ...
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1answer
32 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
22 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: ...
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1answer
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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 ...
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68 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 \\ ...
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3answers
762 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 ...
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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 ...
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1answer
37 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 $2^n - ion$ ...
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85 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 ...
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1answer
54 views

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma}$?

In quantum mechanics we learn about the Pauli spin matrices: $$ \sigma_1 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right) \hspace{0.25in} \sigma_2 = \left( \begin{array}{cc} i ...
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What does it mean that quaternions/ spinors are negated under a full rotation?

As I understand it, quaternions are a type of object called a spinor. Spinors are objects that are negated under a full rotation and only return to their original state under two full rotations. But ...
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345 views

Generalizing complex numbers: Is there a mathematical system isomorphic to 3 dimensional space?

As I understand it, complex numbers: $ax+i$ are isomorphic to two-dimensional space. Quaternions consist of $4$ dimensions. Is that right? Wikipedia says "quaternions form a four-dimensional ...
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1answer
34 views

How is it possible to show that the norm on the elements of Hamilton Quaternions is such that $N(\alpha \beta) = N(\alpha)N(\beta)$?

Let $\alpha,\beta \in \mathbb{H}$ and the norm on $\mathbb{H}$ is defined as $N(\alpha) = \alpha \bar{\alpha}$. How is it possible to show that the norm on the elements of Hamilton Quaternions is ...
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1answer
48 views

Are the quaternions a domain?

I have to give an example of a non-commutative domain that is not a division ring. My first thought was $R = \big\{ a + bi + cj + dk \mid a,b,c,d \in \mathbb{Z} \big\}$ since $R$ is clearly ...
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31 views

Gimbal lock easier to control with quaternions?

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come? They use less memory, and are commutable, and provide an smooth rotation along nonlinear ...
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1answer
25 views

Maximal right ideal of $\mathbb{H}[x]$

Hi I'm trying to prove the right ideal $(x-i)\mathbb{H}[x]$ of $\mathbb{H}[x]$ is maximal. I've tried defining a surjective function $f:\mathbb{H}[x] \to \mathbb{H}$ by $g(x) \mapsto g(i)$ and using ...
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1answer
57 views

Formula for quaternion exponentiation

I'm having trouble understanding the polar representation of quaternions. That is, any quaternion $z=a+ib+jc+kd=a+\mathbf{v}$ can be expressed in polar form as: $$ z = |z|\left(\cos \theta ...
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1answer
45 views

Matrix Algebra over Algebraically Closed Field

In Maclachlan and Reid's The Arithmetic of Hyperbolic 3-Manifolds, when proving that quaternion algebras are simple, they make use of the fact that $M_2(K)$, where $K$ is an algebraically closed ...
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1answer
32 views

Is there a relationship between Rotors and the Rodrigues' rotation formula

I am trying to understand quaternion in general, and it seems like the path to making sense of how they actually work is to first understand rotors and other techniques related to rotations. By ...
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Rotors/Quaternions: double reflection question

I am trying to learn/understand quaternion. I found this reference (among many others): http://www.geometricalgebra.net/quaternions.html It states (see attached screenshot of that page), that to ...
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1answer
42 views

Map from unit quaternions to SO(3)?

On the wikipedia page for "Rotation Group SO(3)" I read that there is a 2:1 surjection from the unit quaternions, $q=w+xi+yj+zk$, to the rotatation matrix $$Q= \left( \begin{array}{ccc} 1-2y^2-2z^2 ...
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39 views

rotate geometry along curve velocity without roll

I am a programmer and I'm writing a script that turns any 3D function into a 3d tube (discrete geometry). In this example I have a bezier curve f that loops and a set of vertex offsets V that ...
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1answer
34 views

How do Quaternions return the Rodriguez formula for rotations?

While trying to work out the general formula for quaternion rotations, I found myself having difficulty getting the result to be the same as the Rodriguez formula as is suggested by multiple works: ...
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1answer
34 views

Why does a quaternion rotation matrix simplify to this?

I'm reading Ken Shoemake's explanation of quaternions in David Eberly's book Game Physics. In it, he defines the rotation matrix for a quaternion $q = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} + ...
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0answers
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How to get the rotation angle about a fixed direction when the object is rotating?

I have posted a question How can I get horizontal rotation angle whatever device orientation? Please see the origin post to get the image of the direction of x, y and z axis. pitch: a pitch is a ...
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2answers
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What is the use of sets above the Complex set?

I recently started reading about sets above the complex set (the set of quaternions, the set of octonions, etc...) and since I already had a lot of difficulty understanding why complex numbers were ...
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650 views

How can Hamilton's quaternion equation be true?

I'm reading Ken Shoemake's explanation of quaternions in David Eberly's book Game Physics. In it, he describes the $\mathbf{i}, \mathbf{j}, \mathbf{k}$ components of quaternions to all equal ...
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1answer
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Given a start point in 3d and a quaternion and length to Point B can you find Point B

Let's assume I have a start point A (x, y, z). Now the object has moved and the new orientation is given by a quaternion Q and it's pointing at point B which is L length away from it. How can I ...
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38 views

Smallest symmetric group with subgroup Q

What is the smallest $n$ such that the quaternion group is a subgroup of $S_n$?
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Questions on Quaternion Algebra (introductory stuff)

I am a relatively new Mathematics student who understands about complex numbers and how they work. I am currently trying to create a 3D computer graphics engine and I heard that quaternion algebra may ...
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Generating set of $\mathcal{Q}_8$

The definition of a generating set $S$ of a subgroup $H\leq G$ is such that if $S$ is a subset of $H$, then for any subgroup $K\le G$ containing $S$, $H\leq K$. Now, for $\mathcal{Q}_8$, if $S=\{i, ...
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How to rotate the origin of rotation of a quaternion

Im working with a quaternion, and its roll pitch and yaw are based on its global location(Im not actually sure how quaternions work, im guessing its on fixed axis). So when i get the values they are ...
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1answer
25 views

Is it safe to say that 3d objects only have 2 and 1/2 rotations

So I've been getting into the math behind animations in video games, specifically Quaternions; and I've noticed that when extracting Euler Angles from a Quaternion, the Yaw is limited from $-90$ to ...
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1answer
53 views

Does the Dirac belt trick work in higher dimensions?

If the Dirac belt is in 4-space, is it still true that when the belt is initially given a 360 degree twist then it cannot be untwisted? I assume this is so because SO(n) is not simply connected, but ...
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1answer
67 views

Can there be a set of numbers, which have properties like those of quaternions, but of dimension 3? [duplicate]

We all know what the complex numbers are : basically $$R^2$$ with a specified product formula, which is $$(a,b).(c,d)=(ac-bd,ad+bc).$$Hamilton defined the quaternions, which are a wonderful set of ...
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2answers
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Prove that the group $H^*$ is Isomorphic to the group $S^3 \times R$?

I'm trying to prove that the quaternion group $H^*$ is isomorphic to the direct product $S^3\times R^+$ where $S^3$ is the 3-sphere which has unit length 1. And $R^+$ being the group of positive real ...
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2answers
23 views

multyplication of 2 vectors forming a matrix - meaning

I am trying understand an algorithm used to determine orientations. Knowing a cross product of 2 vectors gives you a third vector which is orthogonal. What does the multiplication of a 3x1 and 1x3 ...