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

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Calculate $F$ and describe this subset of $\Bbb H$ geometrically.

$F$ is the polynomial function $F:\Bbb H\to\Bbb H$. $Z = \{q \in \Bbb H: F(q) = 0\}$. Calculate $Z$ and describe the subset of $\Bbb H$ geometrically. i) $F(q) = qiq + j$ ii) $F(q) = q^4 - 1$ ...
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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
56 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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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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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
30 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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24 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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35 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
46 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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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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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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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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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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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
38 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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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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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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44 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
29 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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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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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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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 ...
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60 views

Are quaternions used in tensor analysis?

At the moment I am learning tensor analysis. In the books I study (civil engineering/mechanical books on tensors) there is a lot written about rotation matrices/tensors, however quaternions are not ...
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Meaning of symbol similar to $\not >$ in front of a matrix

I found the following symbol in a paper about rotations using quaternions: The paragraph appears at the beginning of page 635 in Closed-form solution of absolute orientation using unit quaternions ...
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1answer
32 views

Show 3D-division algebra over the reals cannot exist using linear algebra

There is a great comment by Jyrki Lahtonen here: Why is quaternion algebra 4d and not 3d? It is not too difficult to show that a 3D-division algebra over the reals cannot exist. If $D$ were such a ...
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23 views

Delta quaternion to euler adjustment

If I currently have two quaternions, $q_1$ and $q_2$, but I can only adjust in Euler values, how would you start to figure out what you need to add or subtract from the $x/y/z$ axis in order to obtain ...
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2answers
111 views

How to calculate the quaternion from/and axis angle having parent and target position (camera and its target)?

I want to calculate the orientation (quaternion) of the virtual 3d camera that is looking at some point in 3d space. The illustration: According to this explanation the quaternion be calculated ...
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Can I write an expression that enters the quaternion space without expressing the variables associated?

First of all, forgive my little knowledge on the subject. I can enter the complex space by just using an expression only having real numbers, for example: ...
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2answers
48 views

Quaternion Multiplication: What is the correct way of doing it?

I am not very familiar with quaternions, I was just doing a programming homework were I had to implement quaternions' arithmetic, however I got puzzled by the multiplication of 2 quaternions. Let's ...
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52 views

Slerp formula interpretation

I have a problem about spherical linear interpolation, or slerp for short. As linked, Wikipedia gives the following formula for an interpolation between quaternions ...
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1answer
86 views

How are Quaternions derived from Complex numbers or Real numbers?

I understand how complex numbers are derived from real numbers. Namely when you have a sqrt of a negative number you must have an answer of some kind, but this answer cannot be in the real number ...
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1answer
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What is the perspective projection of a 3d point relative to a quarternion encoded camera?

I'm representing a camera on the cartesian space as a tuple of a 3d point (position) and a quarternion (rotation). I get the front, right and up vectors of the camera by applying the quaternion to the ...
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Why should quaternions exist?

Why do quaternions exist? I want to believe they exist, but all I can think of are reasons they should not exist. These are my reasons. The quaternions are defined by the following equation: ...
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62 views

Angular Velocity calculation

I am trying to calculate the time derivative of the quaternion from the following paper: Robotics and Biomimetics (ROBIO) See equation 1 below: ...
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1answer
51 views

Calculate hand position from upper-and lower arm's orientation

I've got two unit quaternions in world space representing the lower- and upper arms orientation. The lengths of upper- and lower arm are known. How can i calculate the hand-position relative to the ...
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and ...
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1answer
82 views

Rotation in 4D?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is ...
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1answer
79 views

Expressing unit quaternions in three degrees of freedom

Short version of question: I am trying to use quaternions to avoid gimbal-lock, but I don't know how to express unit quaternions using three degrees of freedom without re-introducing Euler angles and ...
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How to compute angular velocity given a set of unevenly spaced quaternions/direction cosine matrices

I have the time evolution (unevenly spaced) of around 1000 quaternions which provides the transformation from an inertial coordinate system to a body fixed. My goal is to obtain the angular velocity ...
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1answer
117 views

Quaternion rotation intuition

Say the quaternions real and imaginary part are written as $(q_1, \vec q)$. One useful multiplication property is $qr=(q_1r_1 - \langle\vec q, \vec r\rangle, q_1\vec r + r_1\vec q + \vec q \times \vec ...
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1answer
66 views

Does symplectic K-theory $KSp$ have products?

The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the ...
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Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
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2answers
95 views

Concise description of why rotation quaternions use half the angle

I'm currently writing the report on my master thesis project, where I use Android sensors to perform inertial navigation in a heavy industrial environment. In my application, I make use of quaternions ...
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1answer
49 views

Could someone explain the notation of the average of quaternions equation?

The equation has some notation that is difficult to find the meaning for. It is equation (3) in the paper 'Quaternion Averaging' by F. Landis Markley, et al. on page 3 under 'The Average Quaternion'. ...
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1answer
50 views

What does the subspace of SO(3) corresponding to zero yaw look like

Background : I'm solving an engineering problem where I have to estimate the orientation of a body in 3D space. Usually, I use quaternions to do this, but I have to consider a special case where I ...
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1answer
115 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by John Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic ...
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Constructing a periodic function around $f(t): R \to R^3$

Definitions: $f(t): R \to R^3$ $\hat{\bigtriangledown}f(t) := \frac{df(t)}{dt} \frac{1}{|\frac{df(t)}{dt}|}$ $\vec{a}e^{\vec{b}x} := \vec{a}\cos(x) + \vec{b}\sin(x)$ $\vec{a}, \vec{b} \in H$ Is ...
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1answer
60 views

Quaternions and rotation

Basically, I am programming an iOS application where I use attitude of the device in quaternion format. Problem is following: Practically: I have a device that does a measurement #1 of magnetic ...
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2answers
120 views

Clarification of definition of “inverse” with quaternions

From what I understand, the inverse of a matrix only exists if the matrix is square. I recently learned however that the inverse of a quaternion is the quaternion vector (1xn dimensions) where each ...
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3answers
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Multiplication of quaternion vectors

Upon watching a lecture on quaternions (Youtube link), I came across the following math: $$(a,\vec{v})(a,- \vec{v})=(a^2+(\vec{v}\cdot \vec{v}),-a\vec{v}+a\vec{v}+(\vec{v}\times \vec{v}))$$ where $a$ ...