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

learn more… | top users | synonyms

0
votes
0answers
21 views

Constructing a periodic function around $f(t): R \to R^3$

Definitions: $f(t): R \to R^3$ $\hat{\bigtriangledown}f(t) := \frac{df(t)}{dt} \frac{1}{|\frac{df(t)}{dt}|}$ $\vec{a}e^{\vec{b}x} := \vec{a}\cos(x) + \vec{b}\sin(x)$ $\vec{a}, \vec{b} \in H$ Is ...
0
votes
1answer
47 views

Quaternions and rotation

Basically, I am programming an iOS application where I use attitude of the device in quaternion format. Problem is following: Practically: I have a device that does a measurement #1 of magnetic ...
0
votes
2answers
55 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 ...
1
vote
3answers
42 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 $a$ ...
2
votes
4answers
76 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
1answer
41 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 ...
0
votes
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 ...
0
votes
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!
0
votes
0answers
16 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}) ...
0
votes
1answer
24 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'} = ...
1
vote
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 ...
2
votes
1answer
62 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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
1answer
45 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 ...
0
votes
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 ...
1
vote
1answer
27 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 ...
0
votes
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 ...
7
votes
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 ...
1
vote
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 $q$? ...
1
vote
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 ...
0
votes
0answers
75 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 ...
8
votes
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 ...
2
votes
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
votes
1answer
40 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 ...
0
votes
0answers
34 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 } ...
3
votes
1answer
69 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\} = ...
0
votes
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 ...
0
votes
0answers
57 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 : ...
0
votes
0answers
25 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
0answers
47 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 ...
1
vote
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} ...
0
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
44 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 ...
4
votes
2answers
111 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 ...
2
votes
2answers
79 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 ...