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

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How can I visualize Quaternion Linear Interpolation?

It’s hard enough to visualize a quaternion, geometrically speaking. A complex number is simple: it’s a point in a plane. Suppose we had a number like this: a + bi + cj I supose you can visualize ...
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28 views

Multiplying quaternions vs multiplying rotation matrices

It's a trivial question, but one I'm not 100% clear about. Given two matrices $$P_{\{1,2\}} = \left[ \begin{array}{cc}R & t \\ \textbf{0} & 1 \end{array}\right]$$ where $R$ is a 3x3 ...
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1answer
51 views

Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Let $f$ be irreducible over $\mathbb{Q}$ with splitting field $F$. Suppose $\text{Gal}(F/\mathbb{Q})$ is either $D_8$ or $Q_8$. What are the possibilities for $\deg f$? I'm using Dummit & Foote,...
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Is it possible a (3x3) matrix (3x1) vector multiplication represent by quaternions?

Nowadays I am studying rotation using quaternion. I understand, that rotation can formulated a several way. In matrix notation: $$ \vec{v}^{new} = \bar{\bar{R}}^{new}_{old}\cdot\vec{v}^{old} $$ where $...
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16 views

Quaternion update expression [closed]

For a rigid body I have the derivative of a quaternion and the angular velocity in body frame. Now I want to update my quaternion, or find the new quaternion after a time dt. Can you provide an ...
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0answers
29 views

How to prove the derivations of quaternions are all inner? [closed]

I am trying to prove the derivations of quaternions $\mathbb{H}$ are all inner, i.e., $\mathfrak{der}(\mathbb{H})=ad(\mathbb{H})$, but I got no ideas about it. The thing is that I was trying to prove ...
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23 views

discrete logarithm with complex numbers

let $z = a + bi$ where $a,b$ are integers on $[0,N)$ let $a + bi \mod t = (a \mod t) + (b \mod t) \cdot i$ Consider the problem of finding $e$ where $z^e \mod N = c$ and $c, N$ and $z$ are known. Is ...
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25 views

Find rotation in arbitrary axis of a quaternion?

In my situation, I have two completely different Quaternions and an arbitrary axis. What I need to find is the difference of rotation around that axis. For example, if both quaternions had the axis (...
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2answers
41 views

Quaternion interpolation in 3D

I'm a chemist lost in the captivating world of mathematics thus if you could keep your answers simple it would be awesome! Here is my problem: I have two mobiles (A,B) in 3D. Ideally, I would like to ...
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1answer
20 views

The meaning of spacecraft attitude represented in quaternion

I am reading the following paper about the attitude control of aircraft: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1271671 The quaternion represents the relative orientation of two ...
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1answer
25 views

Interpretation of the Derivative of a Quaternion

Considering this definition of the derivative of a quaternion: $$dq/dt = 1/2 w q$$ If we're considering $q$ to be a unit quaternion representing an orientation in 3D with $(cos(theta/2), sin(theta/2)*...
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32 views

How to derivative the function which have constraint like $x^2+y^2+z^2 = 1$

For example , we have function like $f(x,y,z) = 1 - 2\times x^2 + y + z$ with constraint $x^2+y^2+z^2 = 1$ when I need to compute the derivative of $\frac{∂f}{∂x}$ , should it be $\frac{∂f}{∂...
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1answer
31 views

Derivative of rotating a time changing vector by a time changing quaternion

I have a quaternion $q(t)$ that is a function of $t$ and a vector $v(t)$ that is a function of $t$ and I rotate the vector by the quaternion: $f(t) = q(t) v(t) q^*(t)$ but now I want to find the ...
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0answers
21 views

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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1answer
46 views

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
19 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 RHS<->...
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2answers
47 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: $3x^2-...
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1answer
22 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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7answers
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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
29 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
19 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
119 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: $$ijk=-1$$ $$ijk\cdot ijk=i^2\cdot j^2\cdot k^2=(-1)(-1)=1$$ $$i^2=-1$$ $$j^2=-1$$ $$k^...
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0answers
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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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0answers
21 views

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
35 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
37 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
11 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
27 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
36 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
46 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
33 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
41 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
46 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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2answers
22 views

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 euler-...
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0answers
60 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 http://...
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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
64 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' = Rp$...
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1answer
42 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
71 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 $$S^3\xrightarrow{f_1\times ...
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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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35 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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35 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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2answers
68 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
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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38 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 ...