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

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1answer
56 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
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33 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
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1answer
18 views

Dot product of of quaternion-rotated vectors

I'm reading http://people.csail.mit.edu/bkph/articles/Quaternions.pdf and it says "it is easy to show that the operation preserves dot-products." on the page 3. But how is it done? I tried to make a ...
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9 views

How to transform Tait-Bryan-Angels to different rotation orders?

I am having trouble finding or understanding how to get Tait-Bryan-Angels from a rotationmatrix. I have a given rotation matrix $R_q$ which was calculated from the quaternion $q$. I know how to ...
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1answer
42 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
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382 views

$4D$ rotations and quaternions

I have a question about $4D$ rotation: I programmed a little $4D$ game and I used the classical hyper-sphere coordinates, to rotate a vector. It works, but it has some problems: (just for clarity I ...
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1answer
115 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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1answer
454 views

How can I transform coordinate systems with quaternions?

I have a coordinate system 0 which I'd first like to rotate about its z-Axis which gives me system 1, and afterwards rotate system 1 about its y-axis which gives me system 2. See picture: Both ...
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26 views

Calculating two rotation angles from xyz coordinates for dummies

This post is a bit verbose so that others who come later may benefit from my thick headedness. I am attempting to construct a primitives composition and constructed solids geometry parser/processor ...
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1answer
19 views

Multiplication of a quaternion and a scalar to produce a vector?

I am looking at someone else's code, and in it they have a quaternion multiplied with a scalar in order to produce a vector. He used the boost library, and can't find exactly where they defined the ...
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29 views

Geometric significance of map, quaternions.

Let $u, v, w \in \mathbb{R}^3$ be a triple of vectors which form an orthonormal basis in $\mathbb{R}^3$ (with the standard orientation). Identify $u, v, w$ with quaternions in the $\mathbb{R}$-linear ...
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Clarify: “$S^0$, $S^1$ and $S^3$ are the only spheres which are also groups”

The zero, one, and three dimensional spheres $S^0$, $S^1$ and $S^3$ are in bijection with the sets $\{a\in \mathbb{K}:|a|=1\}$ for $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$ respectively. The ...
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24 views

Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors. What I would like to know is angles ...
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1answer
38 views

Hasse invariant of quaternions over $\mathbb{Q}_p$

I am trying to compute the Hasse invariant of the quaternion algebra over $\mathbb{Q}_p$. I denote this algebra by $H$, and I'm assuming $p\equiv 3\pmod{4}$. So, $\mathbb{Q}_p(i)$ is an unramified ...
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1answer
63 views

Why multiplying those 2 quaternions doesn't give the expected result?

Using a left-handed coordinate system, let Q = axisAngle({0,0,1}, 1/4*pi) * axisAngle({0,1,0}, 1/4*pi) be the quaternion representing the rotation "1/8 circle ...
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1answer
66 views

prime divisor propertyfor Hurwitz integers

The Hurwitz integers $\mathcal{H}_{\mathbb{Z}}$ is a particular subset of quaternions. Define: $$ \mathcal{H}_{\mathbb{Z}} = \left\{ a\frac{1+i+j+k}{2}+bi+cj+dk \ | \ a,b,c,d \in \mathbb{Z} \right\} = ...
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1answer
44 views

what represents this strange matrix [closed]

But I just encountered a very strange type of matrix. What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on ...
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1answer
30 views

understanding quaternions - spatial rotations

I would like to know if my understanding about quaternions is correct please: lets say you have a vector in 3d space. You could rotate the x,y and-z frame on a fixed point so that it is parallel with ...
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44 views

quaternions - understanding a formula

Quaternions are new for me. I am trying to understand the following formula: What are: $\large{q^x}$ ? I don't think it is a power. $\large{q^t}$ ? just a transposition of the quaternion $q$? ...
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97 views

Eigenvalues of Group Elements and Quaternions

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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427 views

Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
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67 views

How do I get the rotation between two rotationmatrices?

I am stuck on a little rotation problem. The problem: I have 2 rotation matrices $A$ and $B$. $A$ and $B$ are relative to the coordinate system O. $A$ and $B$ are Quaternion rotation matrices. I am ...
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1answer
31 views

Question about Eigenvalues of group elements

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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1answer
33 views

What is the mapping from purely imaginary quaternions to a vector in $\mathbb{R}^3$

It is claimed that $q = x{\bf i} + y{\bf j} + z{\bf k}$ has an one to one mapping to a vector $v \in \mathbb{R}^3$ where $v = x \hat i + y \hat j + z \hat k$ But ${\bf i}, {\bf j},{\bf k}$ are ...
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32 views

Prove $SU(2)$ is isomorphic to the group of quaternions of norm 1

How could I start finding the isomorphism? Intuitively, a quaternion can be expressed as two complex numbers $a+bi+cj+dk=a+bi+(c+di)j$, and as an element of $SU(2)$ is $\left[ \begin{array}{ c c } ...
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46 views

Rotation that is swapping quaternion $w$ and $x$, $y$ and $z$

I have a quaternion, representing a rotation, equals to: $(w, x, y, z)$ If I convert that quaternion to euler angles, add 180 degrees to $x$ and $z$, and convert it into a quaternion again, I get : ...
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0answers
14 views

Homomorphisms in $Q_8$ [duplicate]

Prove directly that the 2-dimensional irreducible representation $\rho$ of $Q_8$ is not realisable over $\mathbb{R}$. Suppose $\rho: Q_8 \rightarrow GL_2(\mathbb{R})$ is a representation with ...
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20 views

Quaternion Rotation and Transform Exercises with Answers

I work for a company that develops a lot of navigation based software. Most of the problems that we solve can be tackled using rotation matrices, but I've been doing some reading recently about the ...
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21 views

Cartesian extremities of a 3d segment

I have a segment in 3d space and I want to calculate its extremities. I know the cartesian coordinates (x,y,z) of the segment's middle point, the segment's length L and the segment's orientation using ...
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19 views

Condition under which Hurwitz quaternion has left or right gcd equal to 1 with its conjugate

Correct me if I am wrong but for Gaussian integer - $a +bi$ its $gcd (a+bi, a-bi) = 1$ - when $gcd (a,b)=1$ and $a +bi \neq 1+i$ I wonder if there are any known conditions under which Hurwitz ...
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33 views

Grothendieck group of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such ...
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2answers
91 views

Can we describe quaternions using bra-ket in quantum mechanics?

For example, the rotation plus translation of a point using the language of quaternions is written as $Q(0,x,y,z)Q^* + T$ where $Q$ is the unit quaternion, $(x,y,z)$ is the point, and $T$ is some ...
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1answer
30 views

THE positive half-spin space of quaternion vector space

I have the following information: $T$ is the one-dimensional quaternion vector space with the canonical action of $\Gamma$, a finite subgroup of SU$(2)$. This makes sense as SU$(2)$ is the unit ...
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1answer
19 views

Correspondence between rotations and pairs of antipodal unit quaternions

I'm having some trouble understanding how rotations of $\mathbb{R}^3$ correspond to antipodal pairs of unit quaternions. In section 1.5 of his Naive Lie Theory, John Stillwell proves the theorem that ...
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36 views

How can I transform coordinate systems based on quaternion data?

I have a single rigid body object, and its orientations in quaternion with respect to two coordinate systems, each is called original and prime, respectively; therefore, I have two quaternions ...
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127 views

Minimum number of real multiplications to multiply two quaternions [closed]

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
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1answer
29 views

Showing two definitions of the Quaternion Algebra are the same

For $q=z+jw$ where $z,w\in\mathbb{C}$, I'm given a map $M:\mathbb{C}^2\rightarrow M_{2\times2}(\mathbb{C})$ given by $$M(q)=\begin{pmatrix} z & \overline{w} \\ -w & \overline{z} ...
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16 views

Eigenvalues of Second Derivatives on Quaternions

So, say you have some quaternion $q = a + b i + c j + d k$. Let $a, b, c, d, \in f(w, x, y, z)$ which each take four real numbers and yield one real number. Then, what are the most general solutions ...
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2answers
51 views

Quaternion order associated to a ternary quadratic form

I am a bit puzzled by the discriminant of a ternary quadratic form. According to Lehman 1992 and another related question, the discriminant of a ternary quadratic form is the half-determinant of its ...
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2answers
93 views

How are quaternions a finite set?

I'm having trouble understanding how Quaternions are a finite set when you can express a quaternion as Q = a + ib + jc+ kd, because a, b, c, d are $\in$ of $\Re$ would this not mean that the set is ...
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1answer
62 views

Can closure of quaternions under multiplication be shown with a cayley table?

Unsure about my understanding of groups and quaternions. I'm trying to figure out if just using a cayley table (specifically this one) can show closure of quaternions under multiplication, is ...
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What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset ...
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3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
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1answer
51 views

Question about $4\times4$ matrix representation of a quaternion

I have a problem to solve about showing the real quaternion group $\mathbb{H}$ is isomorphic to $M_4(\mathbb{R})$ When trying to define my map I was having trouble coming up with an appropriate map ...
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1answer
39 views

3 rotation values to work out rotation in degrees

I am currently working with the Oculus headset and dealing with the Z axis. With the software I have, the values I can retrieve are limited and I was hoping someone could help me find a solution to ...
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2answers
104 views

Quaternion ^ Quaternion [duplicate]

I was looking at Quaternions at Wikipedia - I was trying to find the value of $i^j$ etc... Wikipedia lists $q^\alpha$ where $\alpha$ is real, but I can't find the value of $i^j$. Any clues? The ...
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2answers
72 views

How would I apply an Exponential Moving Average to Quaternions?

I'm trying to filter positional and rotational data using an Exponential Moving Average (EMA) filter. This has worked without issues for positional data (3D vectors) but I can't figure it out for ...
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2answers
62 views

Quaternion algebra of characteristic 2?

I've been reading up on quaternion algebras recently and noticed the vast majority of theorems are contingent on setting the characteristic $p \neq 2$. In particular, this is true for the central ...
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31 views

Euler -> Quat: Flipped: 90 == -270

i am playing around with quaterions, matrices, euler rotations. For some reason when converting from euler to quaternion to euler my angles are flipped. So where i expect 90, i get -270. 30 stays ...
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2answers
22 views

Commutativity of Spatial Rotations

I know that in general spatial rotations (rotations in $\Bbb R^3$) are not commutative. But what if we restricted our possible rotations to only those around orthogonal axes? For instance, what if ...