Questions tagged [quaternions]
For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.
1,562
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Geodesic of quaternion swings vs vector angle
Context
I work on calculating distance between quaternions, the idea at the end is to train a machine learning algorithm to estimate a rotation in quaternion.
I have two quaternions $Q_1$ and $Q_2$ ...
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Numeric system where iterated square roots of 1 have integer parts
Let us define $s(n)$ as any solution to the equation $x^2 = n$, such that $x \neq n$. I am looking for a numeric system such that $s(s(s\cdots s(1)))$ is always an integer or is composed of integer ...
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Unitary groups and special unitary groups
I'm trying to learn about group theory (I'm neofite in this field) for physics reading a book, "Physics from symmetry" by Schwichtenberg. In section 3.2 the author introduces rotations in ...
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Basis of the matrix representation of a quaternion
I'm trying to learn some group theory for physics as a self-learner. I'm reading about rotation in 2D and 3D space. The book I'm following, Physics from symmetry by Schwichtenberg, firstly proves that ...
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Commutate a Quaternion multiplication
I have a fixed Quaternion t that is defined as Quaternion([0.5, -0.5, -0.5, 0.5]);. The order here is ...
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Show that this optimization problem has a closed form solution or at least it's convex
Let $(q_i)_{i\leq n}$ be unit quaternions, $a_{ij}$ be unit quaternions, and $w_{ij}$ be positive real numbers. We consider the expression:$$\sum_{i,j} w_{ij} \arccos(\operatorname{Re}(q_i a_{ij} \bar{...
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find quaternion to rotate a point towards another point in 3d space
I have a point in a 3D space rotated by a quaternion. The point has a fixed length from the rotation itself (shown in image). What I'm trying to achieve is to rotate my point towards a goal position ...
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Every order in a finite-dimensional $\mathbb{Q}$-algebra is contained in a maximal order
Definition. A lattice in a finite-dimensional $\mathbb{Q}$-algebra $V$ is a finitely generated $\mathbb{Z}$-submodule $\mathcal{L} \subset V$ such that $\mathcal{L}\mathbb{Q}=V$ (i.e., $\mathcal{L}$ ...
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Is there any practical use for octonions? [closed]
Quaternions are useful for describing orientation/ rotations in 3- dimensions, however is there much practical use for an 8-dimensional base hyper complex number id est Octonions?
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How can I interpret quaternion orientation into compass heading angles?
I am using a BNO055 sensor to generate quaternion orientation data. My goal is to calculate accurate compass heading angles from quaternion orientation data.
Interpreting compass headings using Euler ...
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Orders in quaternion algebras
Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$.
Throughout, we fix $F=\mathbb{Q}$.
...
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Two dimensional representation of $Q_8$
I am trying to find what this two-dimensional representation of $Q_8$ is. I have deduced its existence from the character table but now I’m trying to actually see how it works.
I guess I’m looking ...
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Ideals of the Lipschitz quaternions
Consider the subring $\mathcal{O}:=\mathbb{Z}+\mathbb{Z}i+\mathbb{Z}j+\mathbb{Z}k$ of the ring $\mathbb{H}=\mathbb{R}+\mathbb{R}i+\mathbb{R}j+\mathbb{R}k$ of real Hamiltonians. Let $I$ be a right ...
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Is this double exponential smoothing algorithm valid?
I use this double smoothed prediction algorithm:
https://cs.brown.edu/people/jlaviola/pubs/kfvsexp_final_laviola.pdf
Equations:
$$
First Smoothing Statistic:\;Sy_t' = \alpha y_t + (1-\alpha)...
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Is it possible to multiply a quaternion by a $3\times 3$ matrix? [closed]
I have a rotation stored in a quaternion $\textbf{q}$ and an inertia tensor of a rigidbody $\mathit{I_b}$ ($b$ for body) stored in a $3\times 3$ matrix.
I would like to express my inertia tensor $\...
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Quaternion division rings in characteristic $0$
Hamilton's original quaternions $\mathcal{Q}$ form a division ring which is $2$-dimensional over $\mathbb{C}$, and which has $\mathbb{R}$ as its center. Define its elements as $a + bi + cj + dk$, ...
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Distribution of distances between random points on spheres
This is in relation to the following blog post by John Baez:
Random Points on a Sphere (Part 1)
It is claimed that, for two unit quaternions $x,y\in S^3$, uniformly randomly selected, the probability ...
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Functions without complex roots, but with quaternion roots
Many introductions to complex numbers begin with the question "What are the roots of $x^2 + 1 = 0$?"
This function does not have real roots, but does have complex roots.
Are there functions ...
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Derivation of quaternion logarithm
I'm trying to understand how the inverse of the quaternion exponential was derived. Given the definition of the quaternion exponential, $$e^Q=e^{a+bi+cj+dk}=e^{a+v} = x+yi+zj+wk = e^a\frac{v}{|v|}\sin(...
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What surfaces can you define can as the image of a quaternion polynomial?
I noticed the other day that you can compute the conjugate and extract the "high" and "low" complex parts of a quaternion using linear functions. This made we wonder what surfaces ...
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Quaternion - calibration sensor [closed]
I have attached my sensor to a box. What I am interested in is the rotation of the box with respect to the world. In my case, to avoid the gimball lock, and beacause the sensor already provides it, I ...
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Cardinality of conjugation of indefinite quaternion algebra over $\mathbb{Q}$ [closed]
Suppose $D$ is an indefinite quaternion (division) algebra over $\mathbb{Q}$. For $a,b\in D$, we say $a$ and $b$ are conjugate if and only if there exists an element $d\in D$ such that $a=dbd^{-1}$.
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Compute the Jacobson radical in Quaternion Algebras
Let $B$ be a finite-dimensional $F$-algebra. The Jacobson radical of $B$ is the intersection of all maximal left ideals of $B$.
The following is an exercise from Voight's Quaternion Algebras.
Suppose ...
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Squared multiplication of Quaternion
I am reading a book on the math behind game engine. At the end of the chapter that talks about quaternions they are some exercises.
I need to prove that $||q_1q_2||^2 = ||q_1||^2||q_2||^2$
It should ...
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Function to approximate infinite series $\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n+1)!}$
I'll provide background information at the bottom of the post for those who are curious, but the problem at hand is finding a function to approximate the value of
$$\sum_{n=0}^\infty (-1)^n \frac{x^{...
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Is the computation of quaternions faster and less prone to rounding errors than the computation of attitude matrices?
I am having trouble understanding this, I hope you guys can help:
Is the computation of quaternions faster and less prone to rounding errors than the computation of attitude matrices, due to the ...
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For Ring of Quaternions, does $a + bi +cj + dk = 0$ imply $a,b,c,d = 0$, where $a,b,c,d \in \mathbb R$?
Of course, if $a,b,c,d = 0$, then $a + bi +cj + dk = 0$. But having trouble with other direction.
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Voight's Quaternion Algebras, Corollary 7.1.2
I am reading the proof that if $B$ is a central simple $F$-algebra of dimension $4$ ($\operatorname{char}(F) \neq 2$), then $B$ is a quaternion algebra in John Voight's book Quaternion Algebras (...
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How to apply a delta Euler angle after a set of Euler angles while maintaining revolutions?
I'm not a mathematician so I will try to explain this as best I could.
I have an application where transforms store rotations $R$ using Euler angles $Rx$, $Ry$ and $Rz$. The user is able to repeatedly ...
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Geometric understanding of "Pauli vector" determinant?
If we define the Pauli matrices as
$$\sigma_0 =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}\quad
\sigma_1 =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}\quad
\sigma_2 =
\begin{...
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Intuition regarding Hurwitz Primes
Let $\mathcal{H}$ denote the non-commutative ring of Hurwitz quaternions. It is stated in the work of Conway and Smith that $\pi \in \mathcal{H}$ is a prime element if the norm of $\pi$ is a rational ...
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Determine the Automorphism group of the quaternion group. [duplicate]
Question: Determine ${\rm Aut}(Q_{8})$ for $Q_{8}$ being the quaternion group.
I'm currently self studying some abstract algebra and I am struggling a little bit with this question from Micheal Artin'...
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Why is the determinant of a matrix representing a quaternion related to its norm?
A common (complex $2\times 2$) matrix representation of the quaternions is given by $\phi(a+bi+cj+dk)=aI+bi\sigma_1+ci\sigma_2+di\sigma_3$, where $\sigma_i$ are the Pauli matrices, as shown here. In ...
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How to construct a quaternion from two or three normal vectors that have direction and torque/roll of a solid body taken from a quaternion
I am working with magnetic coils normal to each other (two or three) that a third party produces a quaternion that captures the direction and the roll around the direction. I must low pass filter the ...
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Given an angle a and vector v, what is the quaternion that represents that 3d rotation?
And vice-versa, given a quaternion, what is the angle and vector that represents that rotation?
I've read elsewhere that a vector can be converted to a quaternion by ...
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why are negative quaternions the same as positive quaternions?
From what I understand, quaternions are a way to represent a rotation
In this formula, n is the axis of rotation and theta is the angle. So if I'm trying to represent the following rotation
The ...
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How to convert Quaternions into Polar form?
I would like to know how to write quaternions as polar form. Because I heard that if $A$ and $B$ are elements of $C$, this can be done with the form $A \cdot e^{B \cdot j}$.
But how can I do that?
Can ...
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a tetraquaternionic magma
In Clifford algebras over $\mathbb R$ you look at directions squaring to $-1$ or $+1$.
Made me wonder: Why does not nature encode source information yet another way: $(\sqrt i)^2=i$ so that $i^4=j^4=-...
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Quaternion derivative proof
In this paper I tried to understand the Quaternion derivative formula derivation for a body angular rates but I got lost right after equation 12 it states that p(t) is a vector since it's scalar part ...
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Dual quaternion norm expression from Skinning with Dual Quaternions Paper.
I am familarizing with a common skinning techinque used in animation, the Dual Quaternion Skinning. Since the original paper is not long I am going through the math myself.
There's equation 3 which I ...
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Identifying if 2 motors are "compatible." (no candy wrapping)
Preamble (Trying to describe candywrapping)
Consider a translation and rotation encoded as a motor in $Cl(3, 0, 1)$, also known as PGA.
All rotations in 3D can be considered to be a planar rotation up ...
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Quaternionification isomorphims
In the book Representation of compact Lie Groups of Tammo tom Dieck, chapter II.6, it is explained that if $V$ is a complex vector space and $W$ a quaternionic module, we have the isomorphisms (where ...
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3d rotation representation for multiple turns
Currently I am learning to use quaternions to represent rotations in the 3D world. The task is to make the object rotate for a given angle in certain time. I am now wondering which type of rotation ...
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Algebra structure of $\mathbb{R}[Q_8]$ where $Q_8$ is the quaternion group of order $8$.
Let $Q_8$ be the quaternion group of order $8$. I would like to determine the algebra structure for $\mathbb{R}[Q_8]$.
I think $\mathbb{R}[Q_8] \cong \mathbb{R}^4 \oplus \mathbb{H}$. Maybe a simpler ...
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Using quaternion to get polar rotational components
I have tried multiple times to understand quaternions and have failed miserably. I know how to use it to extract unit vectors for cartesian operations, but not how to use it directly for these types ...
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Why is the norm of a Hurwitz Quaternion always a positive integer?
Let $\mathcal{H}$ denote the Hurwitz Quaternion, i.e the subring of the ring of real quaternions that is defined in the following way:
$$\mathcal{H} = \{ m_0 \zeta + m_1 i + m_2 j + m_3 k \mid m_i \...
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How to properly offset a quaternion by another
I have a quaternion giving me an orientation $Q_1$ of a Cube in space.
I rotate the cube in some way and get a second orientation of the cube $Q_2$
My goal is to get a quaternion $Q_3$ that would tell ...
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isomorphism of quaternion algebras
I has a question reading 'Galois cohomology-Gille-Szamuely'
I has a problem as follows.
Let us define a k-algebra homomorphism $\varphi:(a,b) \to (u^2a,v^2b) $ which assigns $ui$ to $i$ and $vj$ to $...
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Finding the quaternion which projects one vector onto another
I'm currently working on a program that requires specifying a quaternion rotation to point a cylinder in the direction of its velocity vector, i.e. projecting the $z$-axis onto that velocity vector. ...
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Embedding the generalised quaternion group into a general linear group
It's known that there are four non-abelian groups with cyclic subgroup of index $2$. Those groups are the dihedral group $D_{2^n}$, generalised quaternion group $Q_{2^n}$, modular-maximal group $M_{2^...
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