Questions tagged [quaternions]
For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.
1,663
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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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1
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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 ...
- 113
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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 ...
- 619
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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:
$$\mathcal{H} = \{ m_0 \zeta + m_1 i + m_2 j + m_3 k \mid m_i \...
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isomorphism of quaternion algebras
I has a question reading 'Galois cohomology-Gille-Szamuely'
I has a problem as follows.
Let us define a k-algebra homomorphism $\varphi:(a,b) \to (u^2a,v^2b) $ which assigns $ui$ to $i$ and $vj$ to $...
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Finding the quaternion which projects one vector onto another
I'm currently working on a program that requires specifying a quaternion rotation to point a cylinder in the direction of its velocity vector, i.e. projecting the $z$-axis onto that velocity vector. ...
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Embedding the generalised quaternion group into a general linear group
It's known that there are four non-abelian groups with cyclic subgroup of index $2$. Those groups are the dihedral group $D_{2^n}$, generalised quaternion group $Q_{2^n}$, modular-maximal group $M_{2^...
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A division quaternion algebra in which the integral elements don't form a ring
I wish to find a division quaternion algebra $B$ over $\mathbb{Q}$ and elements $\alpha, \beta\in B$ such that $\alpha, \beta$ are integral over $\mathbb{Z}$ but both $\alpha + \beta$ and $\alpha \...
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Jacobian of $R^{-1} v$ with respect to $R \in SO(3)$
According to "A micro Lie theory for state estimation in robotics", the Jacobian $J_R^{R\cdot{}\mathbf{v}} = -\mathbf{R}[\mathbf{v}]_\times$. Here non-bolded R is an element of SO(3) or an ...
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Non-standard complex structures on $\Bbb H\times \Bbb H$ so that multiplication is holomorphic
Let
$$\mu:\Bbb H\times \Bbb H\to \Bbb H, \qquad (x,y)\mapsto x\cdot_{\Bbb H} y$$
denote the product of two quaternions. With the standard identification
$$\Bbb H\cong\Bbb C^2\cong \Bbb R^4, \qquad x_0+...
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Why isn't the crossed product algebra $(K(\sqrt{a}), Gal(K(\sqrt{a})/K, k)$ independent of a?
I am trying to work through an example to understand the correspondence between $H_2(Gal(K(\sqrt{a})/K),K(\sqrt{a}))$ and $Br(K(\sqrt{a}),K)$. Here is my confusion:
Let $E = K(\sqrt{a})$ be a ...
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Why is SLERP not the same as this method for averaging quaternions?
I understand that the weighted average of at least 2 quaternions is as described here in equation 13. To summarise, the steps are:
Take the weighted sum $\mathbf{M} = \sum_i w_i \mathbf{q}_i \mathbf{...
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Functional equation in quaternion algebra
Let $F,G:\mathbb{C}^2\rightarrow\mathbb{H}\otimes\mathbb{R}^4$ be two functions with components $F_0,F_1,F_2,F_{1,2},G_0,G_1,G_2,G_{1,2}:\mathbb{C}^2\rightarrow\mathbb{H}$ with quaternionic values. ...
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Fractional powers of quaternions
I was looking into quaternion mulitplication, but I'm having a hard time understanding how it works when the exponent of, say $j$, is not an integer.
The basic algebraic definitions of quaternions are ...
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Isomorphism between quternian and SU(2) and their homomorphisms to SO(3)
From Kostrikin, A. I. (1982). Introduction to Algebra. Springer-Verlag,
$$\Gamma: \operatorname{SP}(1)\subset\mathbb{H} \to \operatorname{SU}(2)$$
$$a+bi+cj+dk \mapsto
\left(\begin{matrix}
a+bi &...
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Is the quaternion division well defined? [closed]
Given that in quaternions generally $pq^{-1}\ne q^{-1}p$, how can quaternions form a division algebra?
For instance, $( i+2 j)\cdot\frac{1}{5 j+1}=\frac{1}{26} (i+2 j-5 k+10)$ but $\frac{1}{5 j+1}\...
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$Hom_{\mathbb{C}}(\mathbb{H},\mathbb{H})\simeq\mathbb{C}(2)$
I'm studying the book "Spin Geometry" by Lawson and Michelson. In the proof of proposition 4.2 they say (with no proof) that there is this algebra isomorphism
$$Hom_{\mathbb{C}}(\mathbb{H},\...
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How do I extract the rotation angle about an specific axis from a rotation matrix.
The question sounds similar to Angle of rotation around arbitrary axis from matrix but it is not.
I don't want an angle-axis extraction, from a 3×3 rotation matrix ${\rm R}$. I know how to do that by ...
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What are the quaternion algebras over $\mathbb{F}$ for a field $\mathbb{F}$?
I know that the only quaternion algebras over $\mathbb{R}$ are the quaternions and the split-quaternions. What is the characterization of the quaternion algebras over a particular field? Which of ...
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Apply Quaternion Rotation to Vector
Have done lot of googling on this and am overwhelmed by the number of formulas; not a math major, just a developer struggling to understand quaternion rotations:)
...
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1.5 Stillwell's Naive Lie Theory rotation by conjugation theorem. Order of multiplication
In section 1.5 of Naive Lie Theory by John Stillwell the rotation by quaternion $t = \cos\left(\theta\right) + u \cdot \sin\left(\theta\right)$ is defined as conjugation by $t$: $q \mapsto t^{-1} q t$ ...
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Computing an element that generates the quaternion ideal
Let $B$ be a quaternion algebra over $\mathbb{Q}$, let $\mathcal{O} \subseteq B$ be a maximal order, and let $I$ be a left $\mathcal{O}$-ideal. I know that there is an element that generates $I$, i.e.,...
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I have two vectors u and v that both lie on z = 0. Can I find a quaternion axis to for these vectors if the axis must also lie on z?
Vectors u and v both lie on the x-y plane or have a z value of zero. I understand that there is an infinite amount of quaternions that will rotate u to v. Is there a way to find an axis for a ...
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I have two vectors u and v. Can I find a quaternion and the quaternion axis if my rotation around this axis is 180 degrees?
I have the two vectors v and u as (1x3) matrices. I need to rotate vector v around an unknown axis to the point at vector u. My main question is, is there an axis that will satisfy this problem if v ...
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Non-normal Subgroup of Quaternion Product
I needed to show that the $Q_8\times Q_8$ contains a non-normal subgroup where $Q_8$ is the quaternion group.
My example was the subgroup $\{(1,1), (i,k), (-i,-k)\}$.
I got the question wrong because, ...
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Is this a discovery or something known? "You find any Integer solution for pythagorean quadruples, pythagorean quintuple. using quaternions √
If you take a quaternion number lets call it Q (Q=a+bi+cj)) Q² definitely generate a pythagorean quadruple and Q=a+bi+cj+dk will definitely generate pythagorean quintuple
ex: Q=1+1i+1j+k, Q² = (-2)+2i+...
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How to factor a rotation fixing the origin into 2 reflects with quaternion?
In $\mathbb R^4$ a reflect in a hyperplane through the origin $O$ is, $\forall q \in \mathbb R^4$
$$q \mapsto -u\overline{q}u$$
, where $u$ is a unit quaternion.
In $\mathbb R^3$ a rotation fixing the ...
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How to calculate an endomorphism ring for a supersingular elliptic curve
I've read a few books and papers about isogeny-based cryptography and its mathematic but didn't get the idea how to find the endomorphism ring of a supersingular elliptic curve. I know how to do it ...
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What are the correct camera (quaternion) transformations to change the object view
I'm currently working with some camera rotations. The issue is, given a camera in an arbitrary rotation (R) and position (T) in a three dimensional space I want to give the viewer the impression that ...
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Symbol for quaternion multiplication
I came accross different notations for the multiplication between two quaternions, e.g:
\begin{equation}
\mathbf{q}_1 \circ \mathbf{q}_2 \quad \text{or} \quad \mathbf{q}_1 \otimes \mathbf{q}_2
\end{...
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Why is it that we can represent vectors using the even part of the Clifford algebra?
You can represent a vector by a quaternion with no scalar part, and you can also represent the rotation itself as a quaternion. Then the rotation is applied to the vector by conjugation. The ...
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2 dimensional faithful quaternionic irreps of a finite group
I am interested in finite groups with faithful 2 dimensional quaternionic irreps (finite subgroups of $ Sp_2 $).
Given a finite group $ G $ it is easy to find the character table (using GAP say) and ...
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Quaternion slerp trig for solving scalars of vectors
This problem I'm reviewing from a mathematics text for game programming. It involves some trig, and suddenly I realize I'm having difficulty coming to the same conclusion for solving for $k_0$.
From ...
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When is operator conformality preserved under addition and subtraction?
An operator $A$ on an $n$-dimensional real vector space is conformal in case
$$
A^T A = \alpha I \qquad \text{for some} \qquad \alpha \ge 0 ,
$$
and
$$
\mathrm{det} A \ge 0 .
$$
Let $A$ and $B$ be two ...
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Quotient of HP^n by SO(3) = Aut(H)
Let $\mathbb H \mathbb P^n$ denote the quaternionic projective space. The group $SO(3)$ acts on $\mathbb H$ (by automorphisms of $\mathbb R$-algebras) by acting on the imaginary part in the obvious ...
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how to combine 2 rotaion quaternions
if I have 2 quaternions that represent rotation in 2 different axis say one that rotate 30deg around the x and another that rotate 15deg around the y how can I combine them in one quaternion
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Kernel of a ring homomorphism
Currently struggling with showing that the quaternions are isomorphic to a subgroup of $M_2(\mathbb{C})$.
I've defined a map $\phi$ from $\mathbb{R}\langle x,y,z\rangle$ to $M_2(\mathbb{C})$ such that ...
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Octonions not an associative division algebra?
On this wikipedia page, I read
The best-known examples of associative division algebras are the
finite-dimensional real ones (that is, algebras over the field R of
real numbers, which are finite-...
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How to get a rotation quaternion from two vectors?
I have two vectors $\vec a$ and $\vec b$ in 3d space. Both of these vectors has length (magnitude) 1 and begin from the origin, so $\vec a$ can be turned into $\vec b$ by a unit quaternion $q$:
$$\vec ...
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On the definition of scalar multiplication for quaternionic vector spaces, in Simon, Representations of Finite and Compact Groups (Theorem II.6.4)
I'm studying from Simon's book Representations of Finite and Compact Groups, and I'm struggling with how the quaternionic spaces are introduced (please, correct me whenever I'm wrong; and, please, be ...
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How to solve for the rotation matrices required to go between two vectors?
I am trying to understand rotations, and I need to understand what it takes to rotate vectors from a certain position on the unit sphere to another.
Let's say I am on $p=e_x$ and I want to go to $p'=\...
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Solutions of the equation $x^2+1 = 0$
Let's consider the equation $$x^2+1 = 0$$
Now, if we work in $\mathbb{R}$, there are no solutions.
If we work in $\mathbb{C}$, there are two solutions.
Now I have been studying rings and modular ...
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Can I combine quaternions like this?
I have a rigid body in the shape of a right angle with 3 reference points on it.
For each pose of the rigid body I know the 3d coordinates of each of the 3 reference points.
I am trying to find the ...
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Are there multiple ways of converting Quaternions to Euler Angles?
Today, I discovered something about rotation matrices in in the source code of Blender. There is a function, that returns two different euler angles showing the rotation of a given rotation matrix. ...
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