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

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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
69 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
37 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
379 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
55 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 non-...
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1answer
44 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
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 ...
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1answer
72 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 +\mathbf{n}...
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1answer
49 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 field,...
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1answer
45 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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41 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
52 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
43 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
48 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
40 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} + w\...
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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
52 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
696 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 $\sqrt{-...
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1answer
22 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
72 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, j\}...
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1answer
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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
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
65 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
141 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
29 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
31 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
60 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
31 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 \begin{...
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30 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
49 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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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
54 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
102 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 power)...
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2answers
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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
292 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
44 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
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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
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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 eye)...
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1answer
105 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 ...
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0answers
52 views

Modify independent XYZ-rotations in quaternion

I have a problem where I am given an arbitrary unit quaternion, need to separately adjust the angles around the XYZ-axes and finally put everything together into a single unit quaternion. I have ...
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1answer
31 views

Rotate along outside of sphere

Although this is related to programming, I don't want to know the programming end of this, just the math. Based on mouse movement, I want to rotate around the origin and always face it like in Google ...
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0answers
24 views

Unwind quaternion multiplication

I am trying to understand quaterions division. Imagine I have the following equation, where every member is a quaternion: $$Q = (qq_1)(qq_2)...(qq_n)$$ I suppose that, if I maintain the order of ...
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2answers
211 views

2D analog to Unit Quaternions for Rotation

I have been working with 3D rotations for some time now, wth my preferred implementation being realised using unit quaternions - especially from a computational efficiency point of view by avoiding ...
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0answers
44 views

Why Quaternion rotations are smoother than Euler rotation?

I have stumble upon this phrase several times but can't fully understand what it refers to. "Quaternion rotations are smoother than Euler rotations." I understand that the gimbal lock problem with ...