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

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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 ...
3
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1answer
63 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
121 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
22 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
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 ...
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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
46 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
30 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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49 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
91 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
45 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
233 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
75 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
71 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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1answer
92 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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51 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
30 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
154 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
42 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 ...
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0answers
102 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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2answers
63 views

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
42 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
167 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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2answers
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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
61 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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0answers
73 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
133 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
37 views

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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6answers
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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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0answers
65 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
62 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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92 views

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 ...
3
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1answer
155 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
104 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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0answers
12 views

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
160 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 ...
2
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1answer
87 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
121 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
52 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
52 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 ...
4
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1answer
137 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 ...