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

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Getting Tait-Bryan Angles from Quaternion for a Non-Standard, Left-Handed Coordinate System

I am trying to write a autopilot script for Kerbal Space Program, which requires me to do some conversions between Tait-Bryan angles and quaternions. Unfortunately, KSP uses a left-handed coordinate ...
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10 views

Quaternion to Euler angles conversion

I have written the following MATLAB code for transforming Quaternion to Euler angles based on the mathematical formula from wikipedia: ...
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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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1answer
32 views

Rotation about z axis using quaternions

I am working with quaternions and rotation, but I am missing something about how a rotation expressed as a quaternion works. I also discovered that there are different convention for quaternions (JPL, ...
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1answer
19 views

Can you extract the horizontal component of the change of two quaternions?

I receive orientation data as quaternions, and I'm interested in finding the ground-planed component of the change in angle. I know that the arccosine of the dot product of two quaternions gives me ...
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2answers
23 views

Difference between quaternions depends on initial rotation

The difference $\Delta q$ between two quaternions $q1$ and $q2$ can be calculated as $\Delta q = q1\cdot q2^{-1}$, where $^{-1}$ is the quaternion conjugate. When numerically evaluating the ...
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1answer
60 views

Why does rotation by a quaternion require multiplying two times?

Given a vector $p$, to rotate it by a quaternion $q$, we use the formula: $$p' = q p \hat{q}$$ where $\hat{q}$ is the conjugate of $q$. But if we use rotational matrices, then it's just $$p' = ...
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3answers
437 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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1answer
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Comparision of Axis-angle and Euler-Angles contradicting?

I used SpinCalcVis to compare axis-angle against the euler-angles and think the angle signs of both are contradicting. I used q = -1 0 0 0 as input. Using the ...
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2answers
36 views

Quaternion angle - Opengl rendering

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). I am trying to calculate the angle of rotation around all the three axes and Render a 3D cube using opengl to immitate the ...
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Differences between Quaternion integration methods

I've implemented a Quaternion Kalman filter and i have the choice between multiple way to integrate angular velocities. The goal is to predict futur orientation $q^{n+1}$ from current orientation ...
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Why do division algebras always have a number of dimensions which is a power of $2$?

Why do number systems always have a number of dimensions which is a power of $2$? Real numbers: $2^0 = 1$ dimension. Complex numbers: $2^1 = 2$ dimensions. Quaternions: $2^2 = 4$ dimensions. ...
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Quaternion Kalman Filter Process Noise

I'm implementing a extended Kalman filter using quaternions. I've extended this paper to deal with my custom observations. My state space is analogous to the one in the previous paper : $ ...
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38 views

identity for quaternions' group Sp(n)=Sp(2n,C)∩U(2n)

Could you help me for solving this: Let $Sp(n)$ be the group of linear transformations of $H^n$ such that preserve hermitian form $$\sum_{i=1}^n \overline{q_i}r_i,$$ that $H$ is the quaternions _ the ...
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1answer
28 views

Sextonion Cayley Table

I've been reading up on the sextonions and was wondering if it would be possible to construct a Cayley table for the split sextonions the same way as one would do so for the split quaternions and ...
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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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Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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1answer
34 views

Can we characterize the Möbius transformations that maps the unit sphere onto itself?

Related: Can we characterize the Möbius transformations that maps the unit circle into itself? The Mobius transformation is of the form $$f(z)=\frac{az+b}{cz+d}$$ In the 3D case, all the ...
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53 views

The general relation between Automorphisms and Derivations

My question is about how one would derive a derivation from a given automorphism (or vice versa) of an algebra $A$ (forgive me if I've worded this incorrectly). For example, as explained here ...
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1answer
34 views

How to decompose a unit quaternion into 3 Tait-Bryan quaternions instead of 3 real numbers?

I'm familiar with the formulas for decomposing a unit quaternion $Q$ into chained Tait-Bryan angles $\phi\theta\psi$ (Wikipedia has the formulas for the $zyx$ chain here), but I'm looking to instead ...
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1answer
105 views

General Linear Group over the quaternions is a topological group

How to show that General Linear Group over the quaternions is a a topological group?
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3answers
492 views

Does quaternion multiplication relate to Minkowski space?

A quaternion notated as $a+bi+cj+dk$ can also be written in terms of a scalar and a vector $(a,v)$, where $v$ is the three-vector $(b,c,d)$. In this notation, the real part of the product $(p,q)(r,s)$ ...
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quaternion conversion extrinsic to intrinsic

I Have a system which is supplying me with quaternions, working in opengl I am setting the orientation of a model to the quaternion I am given, and it is seen that all changes in pitch are shown as ...
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1answer
32 views

Calculating a quaternion that represents a given rotation

This is the first time I'm attempting to do a quaternion and I am not quite getting the concept. This is part of a 3 calculation homework question The initial question is Given a 3-D point at ...
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1answer
33 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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What algebra do you get if you switch the sign of one pair of anticommuting quaternion products?

What are the properties of an altered quaternion algebra defined by: ii = jj = kk = -1, ij = -ji = -k, ik = -ki = +j, jk = -kj = +i, Is it associated with any manifold?
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Degree of quaternion product composed with two maps of $S^3$

Let $S^3$ denote the unit quaternions with multiplication $\mu:S^3\times S^3\rightarrow S^3$.Show that if $f_1,f_2:S^3\rightarrow S^3$ are given maps,that the composition ...
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Relative positioning using quaternions

Say I have quaternion $q_1$, which I have achieved from my IMU module. I want to state that current position is $initial$. Then I want to compute Euler angles relative to this initial position at the ...
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2answers
63 views

Quaternion for beginner

QUATERNION ROTATIONI have 2 points in 3d space ,point A= <5,3,6> and B=<8,2,3> I want to rotate "point B" by 30 degree from point A. how do I solve this question using quaternion. Plz, ...
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22 views

Extract the angle of rotation from a unit quaternion

Sorry for boring you my friends before the spring vacation. I am haunted by a simple problem of how to extract rotation angle from a unit quaternion. Suppose $a$ is a unit quaternion which takes the ...
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20 views

Quaternions and Rotations [duplicate]

I have 2 points in 3d space ,point A= <5,3,6> and B=<8,2,3> I want to rotate "point B" by 30 degree from point A. how do I solve this question using quaternion. Plz, explain the steps....
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Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
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Calculate ψ knowing object orientation in 3D through forward and up vector

I've got a so called right, up, forward tridimensional reference plane and an object $P$ in it. Its orientation in space is defined by two vectors, forward and up: -forward gives azimuth $θ$ and ...
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3answers
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How do you rotate a vector by a unit quaternion?

Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
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1answer
37 views

Every Hamiltonian group contains a subgroup isomorphic to $Q_8$

I read somewhere that every Hamiltonian group (a non abelian group with every subgroup normal) contains a subgroup isomorphic to quaternion group. But I cannot find its proof anywhere on net or in ...
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1answer
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Book(s) on Algebras (Quaternions)?

Well, lately I've been looking for a book on quaternions but I've realized that quaternions are a particular case of the named Algebras(I think Geometric Algebra). Since here, I've found all kind of ...
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Translation offset when converting Matrix to Dual Quaternion skinning

i have a problem with dual quaternion skinning. if i convert my matrixes to dual quaternions i have a fixed offset from the bones (rotation is correct). if i transform everything with identity, than ...
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1answer
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Normalizing a quaternion

How do I normalize a quaternion $$q=w + \mathbf ix + \mathbf jy + \mathbf kz = a + v$$ ? I already know: The normalized quaternion is called unit quaternion and can be calculated in this way: $$U_q = ...
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1answer
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Quaternion exponential problem

I have problem with Euler´s form of quaternion. My quaternion $q=\frac{1}{\sqrt{2}}i+\frac{1}{\sqrt{2}}j,$ so $q^2=-1$, because ...
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5answers
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Can Euler's identity be extended to quaternions?

Euler's identity is $e^{i \pi} + 1 = 0$, a special case of the Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, where $\theta = \pi$ (half-turn of the unit circle). It is commonly described ...
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1answer
28 views

Quaternion from global space to local space

I've searched but have not found a response for this question specifically. I have a smartphone with a sensor that gives me a quaternion representing its absolute rotation relatively to the following ...
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2answers
1k views

Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset ...
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1answer
43 views

Why do two different quaternions appear to have the same rotation?

When using a Quaternions I've noticed something I don't quite understand. If I'm rotation $\frac{\pi}{2}$ radians on the Y axis it goes from $[0,0,0,1]$ to $[0,\sqrt{2},0,\sqrt{2}]$. A rotation of ...
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1answer
44 views

difference between 2 quaternions

I'm trying to calculate quaternions relative to a given orientation. It is easiest for me to explain my intentions by means of an example: Suppose you have a vector $v1=[0,0,1]$ and I want to rotate ...
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1answer
430 views

Converting quaternion or $4\times 4$ matrix to $3\times 3$ matrix representation.

I'm working on some code that manipulates an Axis-Aligned Bounding Box (AABB), so it always encompasses the object it borders. I use a $3\times 3$ matrix to re-size the box when it rotates. The ...
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1answer
34 views

Proof of quaternion algebra being simple using norm

I was wondering if the following simple (pun unintended) proof of the quaternion algebra $A=\left(\frac{a,b}{F}\right)$ being simple is valid. I saw many more complicated proofs online, eg: Proof ...
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2answers
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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
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Understanding rotations of $\mathbb{R}^4$ and pairs of quaternions, showing a rotation is a product of reflections in hyperplanes

I am working through Stillwell's "Naive Lie Theory" and am completely stumped by the questions in this section. An example of one of the questions is Show that the rotation that sends $1$ to $i$, ...
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Check that $u_4\bar{u_3}u_2\bar{u_1}=i$ and $\bar{u_1}u_2\bar{u_3}u_4=1$ so the product of the four reflections is indeed $q \to iq$

This is an exercise from "Naive Lie Theory" and $u_1, u_2, ...,u_4$ are the unit quaternions. I have read the section many times but still don't understand. Can someone explain the material and solve ...
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1answer
1k views

How can I transform coordinate systems with quaternions?

I have a coordinate system $0$ which I'd first like to rotate about its $z$-axis which gives me system $1$, and afterwards rotate system $1$ about its $y$-axis which gives me system $2$. See picture: ...