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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2
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1answer
19 views

the angle between two quaternion vectors is equal to half the angle between their corresponding 3D orientations

I have read in several sources, such as this Math Stack Exchange answer and these slides from a computer animation class (p. 44), that given two quaternions $\mathbf{q}_1, \mathbf{q}_2 \in \mathbb{R}^...
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2answers
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If unit quaternions $q_1,q_2$ satisfy $q_1^2=q_2^2=-1$, then the map $S^3\to S^3$, $x\mapsto q_1xq_2^{-1}$ has a fixed point

Let $\Bbb H$ denote the quaternion algebra. We can identify $S^3$ as the subset of unit quaternions. For fixed $q_1,q_2\in S^3$, consider the map $\phi:S^3\to S^3$ defined by $\phi(x)=q_1xq_2^{-1}$. ...
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+50

Verifying algebraic construction of regular 4-polytopes

One nice way to construct the 24-cell and 120-cell is to take a tetrahedron or icosahedron, look its rotational symmetries (of order 12 or 60, respectively) in $\textrm{SO}(3)$, and pull them back ...
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Generalizing Contour Integration to Quaternions

I have recently entertained the possibility of defining complex contour integration for the quaternions. I am somewhat aware that the Frobenius theorem dictates that no division algebra can exist in $\...
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37 views

Compute adjusted frame of reference for quaternion

My question is an extension to a previous question I posed on stack : Measure change in angle during rotation of quaternion . While there is a split between robotics and programming, my question is ...
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1answer
37 views

What is the relation between order in a number field and order in a quaternion algebra?

The study of number field theory and quaternion algebra theory confuses me a lot: there are always defined symmetrical quantities (for example the trace, the norm, the orders). But I can't understand ...
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Quaternion and vectors for a project [closed]

I need to learn all about Quaternions to realize a project. I'm zero on this, but I have knowledge of linear algebra. I need to find resources on this subject, can you help?
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1answer
30 views

Quaternion and Euler angle conversions: $f(q + \delta q) = f(q) +$ what? [closed]

Say we have function $f$ that converts quaternion to Euler angle ($f(q)= \theta)$. I want to know function p where: $f(q + \Delta q) = f(q) + p(\Delta q)$ where $\Delta q$ is a quaternion difference. ...
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Obtain coordinates after rotation and displacement

I have xyz coordinates, a focal value and a quaternion. I was able to calculate angles in different ways from quaternions, but I'm unable of calculate x'y'z' position after rotation and displacement. ...
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1answer
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Quaternion sandwich simplification gives me a term too many

Context: I'm reading Foundations of Game Engine Development by Eric Lengyel. On the topic of rotations with quaternions it is explained that a vector v can be rotated by being "sandwiched" ...
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4answers
181 views

Four dimensional field over complex numbers

A guy in Facebook claims he's come up with an algebraic field extension to the complex plane. He's defined the unit multiplications as $i^2=-1$, $j^2=i$ and $k^2=-i$. This implies that $ij=ji=k$, $ik=...
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Singularity of swing-twist decomposition

I am implementing a swing-twist decomposition of unit quaternions based on the algorithm in SWING-TWIST DECOMPOSITION IN CLIFFORD ALGEBRA. In AppendixA. Which decompositions are impossible there ...
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1answer
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Has someone ever tried to extend the complex number to $i^x=-1$?

I recently watched a series of videos about the history of complex numbers, and as I got to the part about their geometrical representation, I started to get more curious about the square in $i^2=-1$. ...
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4answers
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Showing $a^2=b^2$ in a group with 8 elements isomorphic to Quaternion group

I am working in a group $G$ with 8 elements such that $a, b\in G$ and $ba=a^3b$. Moreover, I am given that both $b$ and $a$ have order 4 and $\langle a \rangle \neq \langle b \rangle$. I understand ...
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2answers
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Quaternions +Geometric (Clifford) Algebra: What Is the Proper Prerequisite Sequence Before Learning These Subjects

What is the systematic prerequisite sequences of learning that must be mastered before approaching the subject of learning Quaternions, and then Clifford Algebra? My ultimate goal is to learn and ...
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2answers
108 views

Measure change in angle during rotation of quaternion

I am observing the rotation of a box. While the rotation is continuous, I've captured the following three quaternions $q_0, q_1, q_2$ to understand its rotation: \begin{equation} q = [s, x \boldsymbol{...
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0answers
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Is the commutativity and associativity of hypercomplex numbers connected on topology or “flatness” of their space?

Complex numbers are considered circular numbers, and they are commutative and associative. But circle is the last internally flat n-sphere. Quaternions are line n-sphere in higher dimensions, and they ...
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1answer
56 views

how to rotate a vector in 3d space around arbitrary axis

In my case I have two arbitrary vectors (suppose vector AB and CD ) and I am assuming that some rotation operation will happen to vector AB to get it the orientation of vector CD. so by using the ...
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0answers
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Quaternion and Rotation Matrix conversions

I studied computer science at university while ago, we used homogeneous co-ordinates, I understand the basic operations of quaternion up to some level!, The conversion were mentioned but not ...
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3answers
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4D Left Isoclinic Rotations as double rotations

According to this page, 4D left-isoclinic rotation matrices are double rotations of the same angle (and same sign). But in the same page, in the context of quaternions, it is mentioned that left-...
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1answer
25 views

A question about quaternions and reference frames

I have 2 rotations represented by quaternions, quaternions A and B respectively. how would I go about finding the rotation from one refence frame to another, say finding the rotation of quaternion A ...
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1answer
32 views

What does it mean to divide a quaternion by its real part?

In one course a teacher said we were going to normalize a quaternion and did this: $$\require{cancel} q = w + xi + yj + zk $$ $$ q` = \cancelto{1}{\frac{w}{w}} + \frac{x i}{w} + \frac{y j}{w} + \frac{...
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3answers
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A question about quaternion [closed]

I am an 8 grade student.Who can tell me what is quaternion???Please use SIMPLE and understandable words(I am a Chinese)
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1answer
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Unscented (dual) quaternion kalman filter for pose estimation

This paper Unscented Dual Quaternion Particle Filter for SE(3) Estimation shows a method to use Unscented Kalman Filter (UKF) on dual quaternions in the pose estimation problem. It proposes a ...
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1answer
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New coordinate of another point after rotating one point on $S^2$?

As we rotate $p_a = (x_a, y_a, z_a)$ to $p_b = (x_b, y_b, z_b)$ (both are known), how can we find the the new location of $p_c = (x_c, y_c, z_c)$ after this rotation? Can we instead find out the new ...
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1answer
66 views

Rotation angle from quaternion via atan2

I need some help with understanding the rotation angle determination via atan2. The formula which I found in this forum is following: $\phi=2\tan^{−1}(\sin(\phi/2)/\cos(\phi/2))$ This is derived from ...
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0answers
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How to set up the knots for B-splines given degree $k$ and number of control points $n$?

I was trying to replicate how Kim et al converted B-splines onto $S^3$, but the relationships between the knots, the spline degree $k$ and the number of control points $n$ bother me a lot. In ...
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0answers
26 views

I am trying to extract YAW, PITCH and ROLL information from quaternions

I am trying to extract YAW, PITCH and ROLL information from quaternions using angular velocities. this method. This method seems to work if only one axis is non-zero at at a time. Example: when angvx= ...
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0answers
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Use of a-priori known rotation values and Covariance in Manifold Optimization

I've been using manifold optimization to deal with singularities that arise in euler angles in least squares adjustments. Whatever I know comes pretty much from this document “A tutorial on SE(3) ...
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2answers
106 views

How to solve $a^2 - 2b^2 - 3 c^2 + 6d^2 = 1$ over $\mathbb{Z}$?

I would like to find the integral elements int the quaternion algebra $D_{2,3}(\mathbb{Q})$. These are $2 \times 2$ matrices with elements in $\mathbb{Z}[\sqrt{2}]$ with determinant $1$. Observe ...
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Reference for theorem on quotient of generalized quaternion group by its centre being isomorphic to dihedral group

I am looking for a book/paper reference for the following theorem; Suppose $ n \geq 3 $. Let $ Q_{4n} $ be the generalized quaternion group of order $ 4n $. Then $ | Z(Q_{4n} ) | = 2 $ and $ Q_{4n} / ...
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2answers
44 views

Quaternions as rotations: Notation

I am reading through this notes on quaternions. I am trying to understand how they work as rotations as long as their norm is always 1. First, in page 4 above the figure, the quaternion rotation ...
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1answer
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A map $f : sp(1) \to sp(1)$ which commutes with the adjoint representation is a multiple of the identity

Let $f : \operatorname{sp}(1) \to \operatorname{sp}(1)$ be a linear map which commutes with the adjoint representation of $\operatorname{Sp}(1)$, i.e. $f(\operatorname{Ad}_g X) = \operatorname{Ad}_g f(...
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1answer
30 views

differentiation with respect to a quaternion

My question is as follows: if I have a unit quaternion(used to represent the rotation) $q=[q_{0}\,\textbf{q}]$ and a pure quaternion $v=[0\,\textbf{v}]$,where $\textbf{v}$ is a 3x1 vector.* is the ...
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1answer
43 views

What is the derivative of the inverse of the unit quaternion?

the unit quaternion is a good way to represent the rotation. but does anyone know the following formula is right or not?I am trying to prove it by myself but I am stuck on it. $\dot{(q^{-1})}=-q^{-1}*\...
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1answer
33 views

Maximal Tori of $U(n)$ and $Sp(n)$

I am reading Section 3.6 of the Naive Lie Theory by Stillwell. The author claims that we can argue that $T^n$ is the maximal torus of $Sp(n)$ exactly as in the case of $U(n)$. The proof of the maximal ...
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0answers
25 views

Problem with calculating relative orientation

I am using an IMU which provides absolute orientation of the sensors frame $S$ relative to an earth-fixed frame $N$ in quaternion form, $^S_Nq$. In my experiments, I first obtain an initial ...
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1answer
24 views

How to derive rotation matrix for quaternion

I've been following the Wikipedia article on quaternions and spatial rotations and I've come across something I don't understand: . Everything up to this point is clear yet I can't see how the first ...
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37 views

Calculate the rotation angle for a point on a small circle (tilted from XY Plane) to a known point on XY Plane

There are two great circles which are unit circles: Great Circle 1 (GC1): On the X-Y Plane Great Circle 2 (GC2): Tilted 45 degrees about the Y axis Here are some references and constraints: 3D ...
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1answer
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Use quaternion to represent rotation matrix

I know the unit quaternion can represent the 3D rotation. For example, $Rp=q*p*q^{-1}$ where $R$ is the rotation matrix of body frame with respect to inertial frame, $q$ is the unit quaternion, $*$ is ...
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0answers
83 views

Why does not identity theorem works for Quaternions, Octonions, Sedenions etc?

Big chunk of complex analysis is built around the Identity theorem which in layman terms "allows analytical continuation to happen". Is there more or less intuitive way to understand why ...
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1answer
27 views

Reparameterizing quaternion in terms of angle

Why is the red part true? From: http://graphics.stanford.edu/courses/cs348a-17-winter/Papers/quaternion.pdf
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1answer
90 views

Quaternion's multiplicative group as a subgroup of $\mathrm{GL}(4,\mathbb{R})$

Thinking about $\mathbb{C}$: One can view the multiplicative group of complex numbers to be the matrices in $\mathrm{GL}(2,\mathbb{R})$ such that commutes with the matrix $I=\begin{pmatrix} 0 & -1 ...
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66 views

Differential forms in the quaternions

Suppose that $x\in \mathbb{H}$ is a local coordinate chart for $S^4$. Now we have the differential form $dx$, but know I want to compute what is $d\left(\frac{x}{|x|}\right)$ and I am not sure I am ...
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1answer
66 views

Alternative to complex numbers - Study numbers $j^2=1$, $j \neq 1$ [duplicate]

On p. 24 "Clifford Algebra and Spinors" (2.2 Double Ring of $\:{}^2\mathbb{R}$ of $\mathbb{R}$) by Petri Lounesto the author mentions Study numbers as an equivalent alternative to complex ...
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Determine non-unit Quaternions from similarity transformation matrix

Considering a similarity transformation matrix of the form $\mathbf{T} = \begin{pmatrix} m11 &m12 & m13 \\ m21 &m22 & m23\\ m31 &m32 ...
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0answers
21 views

Derivative w.r.t. quaternion for Inverse Kinematics

I'm trying to compute derivative of end effector positions $\mathbf{p}\in \mathbb{R}^{3}$ in Cartesian coordinate with respect to quaterion $\mathbf{q}\in \mathbb{R}^{4}$. When I use euler angles for ...
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1answer
55 views

Geodesic distance of two quaternions

Let $q_0$ and $q_1$ be unit quaternions, and $d(q_0, q_1)$ the angular geodesic distance between the two. I gather that the angular distance can be calculated with the formula: $$d(q_0, q_1) = \lVert \...
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0answers
53 views

Definition of Hecke operators for Shimura curve

I am following Wiles' paper On class groups of imaginary quadratic fields, and it's my first time learning about Shimura curves. There is part of the setup that I don't understand, concerning the ...
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0answers
26 views

I want to write the quaternion $q=2+i+j-5k$ in the form $r+lc$

I want to write the quaternion $q=2+i+j-5k$ in the form $r+lc$, where $r,~c (\neq 0)$ are real numbers and $l^2=-1$. We have the following relation $i^2=j^2=k^2=-1$ and $ij=k,\ jk=i,\ ki=j$. I can't ...

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