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

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Interpolating a vector about an arc (Slerp)

In the following image, how can I solve for $k_0$? I know that $\mathbf v_1$ is a unit vector and $k_1 = \sin tω/\sin ω$.
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Can quaternions be used to represent rotation rate?

A quaternion is a useful tool for representing a rotation, or change in attitude. If a quaternion $q$ represents a rotation, and $v$ a vector, then $v'=qvq^*$ rotates the vector, where the multiply ...
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1answer
15 views

Axes permutations and negations using quaternions

I'm trying to establish conversion between coordinate frames of reference of a phone camera and onboard gyroscope. Because some phones flip Y axis of video, I do not want to limit solution to ...
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2answers
46 views

Is there such a thing as an equation with noncomplex quaternion solutions?

I'm familiar with equations with real solutions and equations with nonreal complex solutions. Examples: $x^2-3x+1=0$ has the real solutions $3\pm \sqrt{5} \over 2$ and this other equation: ...
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1answer
20 views

Estimate angular velocity and acceleration from a sequence of rotations

I have a set of rotations: $R(t) \in R^{3x3}, t = 1, 2, ... T$. I can extract the orientation of a body $\theta (t)$ from the rotation matrix $R(t)$. I am interested to estimate the angular ...
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2answers
28 views

Unit quaternion multiplied by -1

If all components of a unit quaternion (also known as versor) are multiplied by -1, so it still remains a versor, does the resulting versor is considered equivalent to the original versor?
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Why are There No “Triernions”? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
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1answer
28 views

Level set of the hopf map

The Hopf map is given by the projection $\pi: \mathbb{S}^3 \to \mathbb{S}^2$, and: $\pi: z \mapsto zi\bar{z}$, where $z \in \mathbb{S}^3$ and $i \in \mathbb{H}$ is a unit quaternion. Show that ...
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1answer
16 views

How to calculate rotation quaternion between two orientation quaternions?

I have some device (3D pointer) connected to my computer which returns it's position (in cartesian XYZ system) and orientation (in quaternions). I receive this values about 30 times/sec. Now I need ...
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4answers
99 views

How do quaternions not show that $-1=1$? (Where is the proof wrong)

Given the rules of quaternions: $$ i^2=j^2=k^2=ijk=-1$$ could it not be used to show that $-1=1$? As follows: $$ijk=-1$$ $$ijk\cdot ijk=i^2\cdot j^2\cdot k^2=(-1)(-1)=1$$ $$i^2=-1$$ $$j^2=-1$$ ...
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Cauchy-Riemann equation analogue but for the quaternions

given a function over the quaternions $$ U(x,y,z,t)+iV(x,y,z,t)+jW(x,y,z,t)+kR(x,y,z,t)=f(x,y,z,t) $$ what are the analogues of the Cauchy Riemann equation for the quaternionic plane so the function ...
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What do the XYZ and W of a Quaternion represent?

I know that a Quaternion is supposed to represent a rotation around an axis, but I'm still confused as to what exactly do the XYZ and W represent. For example, does X represent the amount I have to ...
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1answer
32 views

how to calculate the result of Quaternion Rotation?

I just read this excellent material page:45 about Quaternion Rotation. I can not compute the result of rotation quaternion $p = [0,\boldsymbol{p}]$ where $\boldsymbol{p}$ is a vector, with $ q = ...
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1answer
31 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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1answer
7 views

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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2answers
26 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
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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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1answer
33 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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2answers
41 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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1answer
31 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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0answers
39 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
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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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2answers
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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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58 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
35 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
63 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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1answer
36 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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2answers
68 views

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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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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34 views

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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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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2answers
64 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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0answers
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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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0answers
37 views

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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0answers
22 views

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
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
36 views

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
62 views

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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0answers
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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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1answer
31 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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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
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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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
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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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1answer
55 views

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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3answers
79 views

Elements of order 2 in the special linear group

I am trying to show that the unique Sylow $2$-subgroup of the special linear group $SL(2,\mathbb{F_{3}})$ is isomorphic to the quaternion group $Q_{8}$. Call the unique Sylow $2$-subgroup $P$, and ...
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0answers
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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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1answer
23 views

Problem with converting rotation representations (quaternion, axis-angle, etc)

I have a computer device - a 3D pointer (Sensable Phantom Omni). It returns cartesian position (X,Y,Z) and orientation quaternion (x,y,z,w). Now I have a 3D visualization software (PyMOL) and I need ...