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

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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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0answers
72 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
776 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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0answers
38 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 ...
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1answer
62 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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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 ...
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1answer
62 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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1answer
31 views

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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1answer
362 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
35 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
51 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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0answers
33 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 ...
3
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1answer
28 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 ...
3
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1answer
67 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
47 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
42 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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0answers
33 views

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
48 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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0answers
41 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
41 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
38 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
47 views

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
51 views

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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3answers
677 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
18 views

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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1answer
43 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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3answers
55 views

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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0answers
27 views

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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1answer
21 views

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 ...
0
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1answer
27 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
59 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
85 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
26 views

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
28 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 ...
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0answers
18 views

Evaluating a Hermite Quaternion Curve

I have a set of fixed poses (position and orientation) and want to interpolate C1 continuous between the orientations. I tried to follow A General Construction Scheme for Unit Quaternion Curves with ...
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1answer
58 views

Proving the direct product $S^3 \times \mathbb R^{+}$ is isomorphic to $H^{*}$

Consider the direct product of the unit 3-sphere with the positive real numbers: $S^3 \times \mathbb R^{+}$ Prove that this group is isomorphic to the non-zero quaternions $H^{*}$ under ...
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1answer
30 views

Matrices made of gluing $\begin{pmatrix} z & iw \\ i \bar w & \bar z \end{pmatrix}$ blocks have a real determinant

Prove that matrices made entirely of blocks of the form $\begin{pmatrix} z & iw \\ i \bar w & \bar z \end{pmatrix}$ have a real determinant. For example, we claim $$\Delta=\det ...
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0answers
29 views

Determining Rotations Applied To A Quaternion From Its Components

Here's a problem I'm seeking a solution to. Step 1: Construct a quaternion, that is coincident with the x-axis. For example, q = (0, 50i, 0j, 0k). Step 2: Construct three unit quaternions, ...
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1answer
42 views

Conversion from euler angles to versors

I am attempting to create a script to convert between the output of one long program and the input of another, neither of which I can edit. The output of the first gives euler angles for rotation and ...
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0answers
27 views

Can a point have a nontrivial isometry group?

This question is extremely related to this other question. In fact, a positive answer here directly implies a positive answer there. However, since it is a mathematically different question I decided ...
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0answers
46 views

Finding the quaternion that performs a rotation

I managed to find this answer here where Christian Rau says "axis/angle rotation (a,x,y,z) is equal to quaternion (cos(a/2),xsin(a/2),ysin(a/2),z*sin(a/2))" Assuming I know what rotation I need to ...
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1answer
81 views

How to calculate sin/cos/tan of a Quaternion?

I would like to learn about Quaternions. I've read this article: https://en.wikipedia.org/wiki/Quaternion Most of the article was not hard to understand, except the (Exponential, logarithm, and ...
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2answers
43 views

Quaternions group $\{\pm 1 , \pm i, \pm j, \pm k\}$ is not isomorphism to Diedral Group $D_4$. [closed]

How to prove that quaternions group $G=\{\pm 1 , \pm i, \pm j, \pm k\}$ is not isomorphism to Diedral Group $D_4$?
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0answers
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Ring Identities

Suppose we had a finite group G with elements $e_0, e_1 ... e_k$ Then consider objects from the set $$ M = { a_0 e_0 + a_1 e_1 + a_2 e_2 ... a_n e_k }, a_i \in \Bbb{R}$$ whereas $$ m + n, (m,n ...
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2answers
203 views

How can i express a quaternion in polar form?

Im trying to express the following quaternion in polar form (axis-angle) $a=1+i-2j+k$ Would this be the resultant ? $$\cos \frac{θ}{2} +\sin \frac{θ}{2} (i-2j+k)$$
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2answers
41 views

Finding the Unit Quaternion

How can i take a Quaternion and find the Unit Quaternion. How can I find the Unit Quaternion (Norm of a Quaternion). The norm of a Quaternion should be equal to $1$ E.g. $a=(2-i+2j-3k)$ Here is what ...
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0answers
66 views

Why are quaternions (over the field of complex numbers) two-rowed matrix algebras?

I've enjoyed D.E. Littlewood's Skeleton Key to Mathematics up to chapter 8. After chapter 8, increasingly more statements are left without proof and I'm lost. He writes [...]it can be shown ...
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1answer
42 views

Quaternion that is not complex

So I have a question which is asking for a Quaternion which is not complex. I'm supposed to find this number on the Internet and we never got introduced to it. Could someobdy give me some kind of hint ...
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0answers
69 views

Is there a residue theorem for Quaternions?

One of Complex Analysis's biggest contributions is the residue theorem. Is there a similar theorem in the field of Quaternion Analysis? (A glance at Wikipedia didn't pull anything that caught my ...
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0answers
21 views

Quaternion to Euler with some properties

I am trying to create a map editor (for GTA SA-MP), and the source game data contains objects with quaternion rotation, whereas I need the editor to output the objects with Euler rotation (XYZ) in ...