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

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2answers
42 views

Clarification of definition of “inverse” with quaternions

From what I understand, the inverse of a matrix only exists if the matrix is square. I recently learned however that the inverse of a quaternion is the quaternion vector (1xn dimensions) where each ...
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3answers
38 views

Multiplication of quaternion vectors

Upon watching a lecture on quaternions (Youtube link), I came across the following math: $$(a,\vec{v})(a,- \vec{v})=(a^2+(\vec{v}\cdot \vec{v}),-a\vec{v}+a\vec{v}+(\vec{v}\times \vec{v}))$$ where $a$ ...
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4answers
75 views

Why is it that with quaternions $ij \neq ji$?

I've been using rotations in 3d space lately for simulations. Today I came across the quaternion, which from what I understand will be a much better alternative to my cross/dot product methods. Now I ...
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2answers
880 views

Rotating a 4 dimensional point?

I'm trying to rotate a 4 dimensional point (w,x,y,z). So far I've been rotating around planes (wx,xy,yz,zw,wy, and xy), but the order in which I do these rotations changes the results and can ...
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1answer
452 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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0answers
25 views

quaternions are less versatile than matrix?

I am doing some research looking should I implement quaternions or matrices. What I've seem to come across is that while quaternions can be better for doing smooth rotations and dual quaternions can ...
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1answer
27 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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1answer
533 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: ...
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0answers
32 views

4x4 Matrix with homogeneous coordinates

I learn for a linear Algebra exam and I have the task: "What is the $4\times 4$ matrix , a rotation about the $\pi/3$ describes in homogeneous coordinates about the axis? What is the ...
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1answer
36 views

why is representing rotations through quaternions more compact and quicker than using matrices??

According to the wikipedia page on Quaternions: The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. However, I have to ...
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1answer
39 views

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$?

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$? This question stems from Stillwell's Naive Lie Theory where he states that a quaternion $t$ of absolute value ...
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1answer
28 views

Equivalance form for Slerp in quaternions interpolation

In all the books I have found that Slerp have two forms: A B I know that all the forms from A are equivalent but I don't know why the forms from A are equivalent with the form from B. Can ...
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2answers
249 views

Exponential Function of Quaternion - Derivation

The equation for the exponential function of a quaternion $q = a + b i + c j + dk$ is supposed to be $$e^{q} = e^a (\cos(\sqrt{b^2+c^2+d^2})+\frac{(b i + c j + dk)}{\sqrt{b^2+c^2+d^2}} ...
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0answers
14 views

Quaternion - trigonometric form - $q=\cos \theta +u \sin \theta$ Components for $u$?

It is proven that a quaternion has the following trigonometric form: $$q=\cos \theta +u \sin \theta.$$ My question is: Which are the components of the $u$? Thanks!
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0answers
15 views

Quaternion and Euler angles small angle proof

Let's start with a quaternion $q = \begin{bmatrix} q1 & q2 & q3 & q4 \end{bmatrix}^T$. Where $q_4$ is the scalar part, which is equal to: \begin{equation} q_4 = cos(\frac{\alpha}{2}) ...
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1answer
23 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
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1answer
49 views

3D Extension of a Fun Geometric Series Puzzle

After being inspired by this question, and in particular Semiclassical's excellent response and generalisation, I thought of another generalisation to a 3-dimensional plane.: Suppose you start at ...
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1answer
70 views

Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)

I am trying to solve the following exercise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. ...
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1answer
60 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
70 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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1answer
24 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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0answers
13 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
43 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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0answers
385 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
127 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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0answers
38 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
26 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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0answers
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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2answers
87 views

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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0answers
28 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
39 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 ...
3
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1answer
67 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
45 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
32 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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0answers
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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1answer
101 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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0answers
431 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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0answers
73 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
33 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 ...
0
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1answer
39 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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0answers
33 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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0answers
56 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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0answers
23 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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0answers
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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0answers
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
108 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
31 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 ...