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

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3
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0answers
22 views

4-D lattices and quaternions

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
0
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0answers
9 views

Angle between Quaternions

I am busy with a project where I have to calculate the angle between 2 quaternions. The inner workings of the project are: There is a reference quaternion and every other quaternion I encounter will ...
0
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0answers
26 views

can somone exaplin how the equations for quanterion roation actually work, or at least show me some?

I don't know enough about qunaterion rotation to as the question easily. Sufficient to say I have a system were three rectangles on a 2 axis plan (I think those words are right.....) that are ...
0
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0answers
21 views

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 ...
0
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0answers
29 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 ...
2
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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,...
-3
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1answer
22 views

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 $...
0
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0answers
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 ...
0
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0answers
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 ...
6
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3answers
188 views

Is there a name for a function whose square is an involution?

An involution is a function $f:X\to X$ such that $f\circ f=\text{id}$. Is there a name for a function $g:X\to X$ such that $f\equiv g\circ g$ is an involution? An example is multiplication by $\pm i$ ...
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0answers
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 (...
3
votes
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 ...
1
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1answer
42 views

Gimbal lock easier to control with quaternions?

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come? They use less memory, and are commutable, and provide an smooth rotation along nonlinear ...
0
votes
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 ...
0
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2answers
55 views

Quaternion and Euler angles small angle proof

Let's start with a quaternion $q = \begin{bmatrix} q1 & q2 & q3 & q4 \end{bmatrix}^T$. Where $q_4$ is the scalar part, which is equal to: \begin{equation} q_4 = cos(\frac{\alpha}{2}) \end{...
0
votes
1answer
214 views

Relative rotation between quaternions

Say I have a quaternion q which describes how to get from frame 0 to frame 1, and a quaternion r which describes how to get from frame 0 to frame 2. To get the "quaternion difference" between q and r, ...
2
votes
1answer
21 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 ...
2
votes
1answer
32 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 ...
0
votes
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)*...
0
votes
0answers
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}{∂...
0
votes
1answer
102 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 ...
0
votes
0answers
22 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 $k_1 = \sin tω/\sin ω$.
1
vote
1answer
20 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<->...
0
votes
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-...
0
votes
1answer
23 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 ...
2
votes
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?
7
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1answer
1k views

Quaternion Rings

Let $R$ be a commutative ring. Define the Hamilton quaternions $H(R)$ over $R$ to be the free $R$-module with basis $\{1, i, j, k\}$, that is, $$H(R)=\{a_0+a_1i+a_2j+a_3k\;\;:\;\;a_l\in R\}.$$ and ...
81
votes
7answers
4k views

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 ...
1
vote
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^...
0
votes
1answer
25 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 ...
0
votes
1answer
20 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 ...
2
votes
1answer
30 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 ...
1
vote
0answers
15 views

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 ...
0
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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 ...
0
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1answer
38 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: ...
2
votes
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 = [\...
0
votes
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-...
1
vote
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 ...
26
votes
6answers
5k views

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: $$i^...
1
vote
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, ...
-1
votes
1answer
26 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 ...
0
votes
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 ...
1
vote
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$...
5
votes
3answers
482 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 ...
0
votes
2answers
48 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 ...
3
votes
0answers
51 views

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 $q^{...
6
votes
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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vote
0answers
50 views

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 : $ \mathbf{...
0
votes
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 ...
0
votes
1answer
34 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 ...