Questions tagged [quaternions]

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

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14 views

Determining attitude from a single set of vectors.

I'm given a large list of absolute points in 3D space, as well as vectors to a subset of these points from a single unknown location, collected from something like a star tracker. I'm tasked with ...
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3answers
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Unit vectors and quaternions

This might be a dumb question but recently I came to know about the quaternion number system. I can't stop wondering, "Are they related to unit vectors in any way? They have similar notations and ...
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Connectedness of non-zero quaternions

I am trying to prove that set of non-zero quaternions is path connected. It is $S^3$, but I am not using path connectedness of higher dimensional spheres (except $S^1$ and $S^2$). I am using the book ...
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1answer
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How to get the left and right rotation quaternions between two $\mathbb{R}^4$ unit vectors?

This question is somewhat similar to this, but in higher dimension. I have two non-collinear unit vectors in $\mathbb{R}^4$, $\bf a$ and $\bf b$. I know from Wikipedia that there is at least one pair ...
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converting a unit dual quaternion to a form using sin/cos and plucker coordinates

Many papers using dual quaternions for rigid transformations state and use the fact that apparently any unit dual quaternion $$\mathbf{q_0}+\epsilon \mathbf{q_\epsilon}$$ can be written as $$\cos\frac{...
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1answer
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Find the matrices of irreducible quaternionic representations of $SU(2)$

The irreducible complex representation of the group $\mathrm{SU}(2)$ of dimension $2n$, $n\geq 1$, is of quaternionic type. In particular, it can be represented by quaternionic matrices $W(g)$, $g \in\...
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Local and global rotation inside hierarchy using quaternions

When I want to rotate a Model using Quaternions I have to either left or right multiply the current orientation with the rotation quaternion: ...
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1answer
34 views

Is there anything like heptonions?

Unit complex numbers can be used to represent roatations in 2D, unit quaternions can be used to represent rotations in 3D. Can there be anything ike heptonions which could represent rotations in 4D or ...
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1answer
45 views

Is ODE factorisation with split-quaternions possible?

In Simple complex ODE's in matrix form? we discussed how we can solve certain complex ODE's in matrix form. As an example: $$y(x)^2+y'(x)^2=0$$ can be factored as $$(y(x)+ iy'(x))(y(x)-iy'(x))=0$$ ...
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1answer
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Can the expression $a^2 + b^2 - c^2$ be factored as a product of two quaternions, where $a$, $b$, $c$ are real numbers?

I'm trying to factor the expression $$a^2+b^2-c^2$$ as a product of two quaternions, where $a,b,c$ are reals. Can anyone give me the answer? I think it can't be done. I started with multiplying $$(Aa,...
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Why does no three-scalar parameterization of 3d orientation exist that doesn't contain singularities?

I'm reading a magazine article on 3d orientation and want to know what the mathematical issue is and where to read about it, preferably a theorem title that I can google or a topic and textbook: It’...
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Expressing common vector operations in terms of quaternions

Since everything we do with vectors could also in theory be expressed using quaternions (or octonions & higher-order -nions), is it possible to have common vector operations like the cross product ...
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First order, non-linear ODE

I am solving the following equation: $$f(x)^2+\frac{f(x)^2}{f'(x)^2}=h(x)$$ for a known function $h(x)\geq 0, x\in \mathbb{R}$. I'm asking for help with: Identifying this ODE in terms of well-...
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Proving that the Octonion Norm Respects Multiplication

I'm working with the following definition of the octonions: $\mathbb{O} = \mathbb{H} \times \mathbb{H}$, endowed with the product $$(p,q)(r,s) = (pr - sq^*, p^*s + rq).$$ Conjugation is given by $(p,q)...
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Quaternion exponent and logarithm to non-standard base

Given that the exponent of a quaternion $q = (w, \vec{v})$ with base $e$ is defined as \begin{align} e^q = e^w (\cos |\vec{v}| + \frac{\vec{v}}{|\vec{v}|} \sin |\vec{v}|) \end{align} and the ...
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1answer
72 views

Quotient of binary icosahedral group by its center, i.e., $2I/\{\pm 1\}$ is isomorphic to $A_5$

I know that the inner automorphisms of the binary icosahedral group is given by the quotient $2I/\{\pm 1\}$ where $\{\pm 1\}$ is the center of $2I$. I also know the elements of the binary icosahedral ...
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$A_5$ or the icosahedral group $I$ is not isomorphic to any subgroup of the binary icosahedral group $2I$

I'm trying to show that the extension $1 \to \{\pm 1\} \to 2I \to I \to 1$ does not split. For that, I think it's sufficient to show that $I$ is not isomorphic to any subgroup of $2I$ (?). But I'm not ...
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Fubini-Study metric of HP^n in affine coordinates

I want to obtain the Fubini-Study metric of $\mathbb{H}P^n$ in affine coordinates, similar to this question but here of the quaternionic projective space. How exactly is the Fubini-Study metric ...
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2answers
63 views

What are the cosets of $\mathbb{Q_8} / Z(\mathbb{Q_8})$?

$\mathbb{Q_8} = \{\pm 1, \pm i, \pm j, \pm k\}$ where 1 is the identity and $\mathbb{Q_8}$ is a non-abelian group under ordinary multiplication Let it be given that $i^2=j^2=k^2=ijk=-1$ I already ...
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Showing that an image of a 3D rotation is a Hopf map

I'm studying geometry and am having trouble with an exercise problem. As a disclaimer, the material is in Korean and there might be some inaccurate things I got wrong when I translated them over to ...
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2answers
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Properties lost when going from real number system to quaternions and octonions

What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?
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1answer
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For unit complex numbers $z$ and $w$ and the quaternion $j$, when does $zj\bar{w}=j$?

I'm trying to figure out for which unit complex $z,w \in S^1$ does $zj\bar{w}=j$ for the quaternion $j$. I was trying to solve this by setting $z=a+bi$ and $w=c+di$, so $zj\bar{w}=(a+bi)j(c-di)=(aj+...
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Find approximating roots to a quaternion polynomial with a naive algorithm

I am trying to devise a Niave Algorithm to find the roots to Quaternion Polynomials. I thought I had found a solution in the article Newton Method in the Context of Quaternion Analysis, as a Newton ...
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1answer
45 views

Do the Hurwitz quaternions form a cyclic group?

I am investigating applications of the Hurwitz quaternions: https://en.wikipedia.org/wiki/Hurwitz_quaternion $$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\...
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1answer
52 views

How would I prove the following theorem on quaternions?

The theorem states The map that takes $q$ to the map $[q] : x \to q^{-1}xq$ is a 2-to-1 homomorphism from the group of unit quaternions to $SO(3)$. How would I prove this?
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114 views

Cannot Find $\beta_2$ in the Following Algorithm

The problem came from a research paper that I'm reading. Please inform me if you need any clarification. So I was implementing the algorithm 1 from literature 1. It aims to reduce a quaternion ideal ...
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History of quaternions representing rotations.

I looked all over, but I cannot find anything about the history of representing rotations by quaternions. Who first came up with this extremely clever idea?
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2answers
79 views

Jacobian matrix of kalman state with quaternion

How can i derive the Jacobian matrix for a Kalman filter state $x$, where $q$ stands for the orientation as quaternion and $\omega$ represents the angular velocity as vector $$x_k= \left[ \begin{...
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Quaternion operation to cancel out roll angle

I'm making experiments with quaternions by using an IMU (BNO055), the sensor can give me orientation as quaternion and euler angles by itself. I'm also practicing with quaternions. My problem is, say ...
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Are there drawbacks replacing quaternions by Conformal Geometric Algebra in an implementation

I am working on a 3D IK engine and stumbled across Conformal Geometric Algebra. I initially did find it a bit confusing but then again quaternions still occasionally perplex me. I am strongly ...
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1answer
42 views

Quaternion PID control

I'm not quite sure how to apply a Proportional-Integral-Derivative control to a quaternion representing orientation. There are a few things I don't understand. Given the PID control formula of: <...
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Calculating body frame angles (Euler angles) between reference frames using quaternion rotation

Problem: Convert Euler angle orientation relative to one reference frame to a new set of Euler angles relative to a new reference frame using quaternion rotation. Example: Consider a given Yaw, Pitch ...
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1answer
20 views

Add a custom extra rotation to a quaternion

I have a quaternion which represents a 3D object rotation, for example: Quaternion(scalar:0.707107, vector:(0, 0, 0.707107)) I want to keep the current 3D object ...
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1answer
82 views

What does the surface |z₁|² = |z₂|² look like?

In a quaternionic plane there are 2 axes and each point corresponds to $q \in \mathbb C^2$. Now, $|z_1|^2 = |z_2|^2$ should define a surface that divides a 3-sphere $|z_1|^2 + |z_2|^2 = 1$ into two ...
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What are “hermitian matrices” and “unitary matrices” called in the context of quaternions?

The real numbers have a conjugation operation $\overline{a} = a$, and we can define the conjugate-transpose operation $A \mapsto A^\dagger$ on matrices. (Here this is just the transpose, because the ...
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Quaternionic quantum channels.

What is the proper definition of quaternionic quantum channel and do we have the choi kraus representation theorem for quaternionic case as well?
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What does the symbol $\stackrel{!}{=}$ in the context of quaternions mean?

I was reading "Physics from Symmetry" by Jakob Schwichtenberg recently. And in part 3 of chapter 3, he goes over quaternions and I found the following statement: The set of unit quaternions $q = a\...
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Understanding the Quaternion rotation identity proof.

I am doing a project about quaternions and their rotation. I am trying to get the proof of quaternion rotation identity by Wikipedia: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation to ...
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To change a quaternion relative to its current state around an axis

Say I have an object with a pose define with the quaternion [0,1,0,0], where the first three values are the vector part. How should I change this quaternion to rotate the pose 10 degree around z axis ...
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1answer
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Quaternion rotationmatrix

I'm a last year student at a high school in Denmark, and I'm trying to understand how quaternions rotations can be described through a rotationmatrix. As of now, I quite don't understand how I go from ...
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1answer
63 views

Show that $\mathbb{H}$ is a division ring

Let $\mathbb{H} \subset M_2(\mathbb{C})$ be the set of matrices of the form: $$A = \begin{pmatrix}z & -\bar{w}\\w & \bar{z}\end{pmatrix} \text{ where } z,w \in \mathbb{C}$$ $A^{-1} = \begin{...
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Normal Vector to Quaternion

I got the following problem: I want to convert the normal vector of a person into an 2D orientation represented as Quaternion. The normal vector is given only in the x-y plane : $$\begin{pmatrix}x &...
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Having difficulty understanding what happens when reversing order of quaternion multiplication.

The textbook I am reading claims that quaternion multiplication works like so: $ q_1q_2 = V_1 \times V_2 + s_1V_2+s_2V_1+s_1s_2-V_1 \cdot V_2 $ Which is a simplified view of $ q_1q_2 = (x_1w_2+...
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Why is a quaternion a + bi + cj + dk not equal to a + (b+c+d)i [closed]

i² = -1 j² = -1 k² = -1 But a + bi + cj + dk is not equal to a + (b+c+d)i why not ?
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Soft question: References for Dedekind's work on quaternions and hypercomplex numbers?

I've been told that quaternions and hypercomplex numbers in general were very important to Dedekind's early as well as late work. Besides Ferreiros' Labyrinth of Thought, ch. 7 and the references ...
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1answer
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Calculate Yaw, Pitch, Roll from Up, Right, Forward

I have a 3D object with a forward, right, and up vector. I'm trying to calculate its Yaw, Pitch and Roll. My last attempt was: ...
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How to define an axis based on multiple quaternions? How to find the angle between two quaternions on an axis?

The project I'm working on is a device to measure the range of motion of a human joint (e.g. the wrist). Inertial measurement units (IMUs) are used to digitally measure rotation. IMUs stream ...
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22 views

Quaternion: Projection from a 3D Plane

I have a 3D Rectangular Slab with position (x,y,z) coordinates of the center and orientation represented by quaternions. I'm trying to get the position of similar rectangular slab parallel to it. How ...
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Does an irreducible quaternionic type representation over $\mathbb{R}$ have a ``good'' matrix expression?

Let $V$ be a real vector space and $G$ be a finite group and $\rho:G \rightarrow GL(V)$ be an irreducible representation over $\mathbb{R}$. It is of quaternionic type if $\text{Hom}_G(\rho,\rho)$, the ...
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168 views

Rotation matrix covariance to quaternion covariance

Given a $3 \times 3$ rotation matrix $\mathbf{R}$ with an associated $3 \times 3$ covariance matrix $\mathbf{P}$ how do I compute the associated $4 \times 4$ covariance matrix $\mathbf{Q}$ of the ...

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