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

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1answer
89 views

Error in Weyl character formula computation.

I need someone with a keen eye for errors. I am trying to use the Weyl character formula for the symplectic group Sp$(4,\mathbb{C})$ on certain matrices coming from 2x2 quaternion matrices. Summing ...
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2answers
808 views

What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
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1answer
83 views

Sub division rings of division rings

Below $^\ast$ denotes "nonzero elements of". There is a problem in Jacobson's Basic Algebra 1, there is a problem to this effect: if $S$ is a subdivision ring of $\mathbb{H}$ such that $S^\ast$ is a ...
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1answer
170 views

Triple quaternion multiplication

I'm self learner and for some reason I can't wrap my head around quaternion multiplication. I just stumble upon one of equation in my text. Can anyone show step-by-step workout for below: $$ ...
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3answers
164 views

How can we make $\mathbb{R}^n$ into a multiplicative group?

Are there any sorts of multiplicative group structures we can put on set $\mathbb{R}^n$? By multiplicative, I mean that it should be compatible with the natural scalar multiplication on ...
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1answer
69 views

How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from…

In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$). ...
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1answer
508 views

3D points rotation to quaternions

For the simplicity, we'll consider two 3D points, that moves one relatively to other, in time. Let's say: ...
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1answer
63 views

quaternion product distributivity

If you check the quaternion product derivation at wikipedia: http://en.wikipedia.org/wiki/Quaternion#Hamilton_product You can see that it is derived from a multiplication table between the ...
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1answer
589 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
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1answer
296 views

Why do Quaternions and octonions exist?

Ok so I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are ...
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2answers
117 views

Correspondence between rotation representations

I was wondering if there is a bijection between unit quaternions and other rotation representations such as vector of rotation, Euler angles or rotation matrices. It seems to me this is not the case ...
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2answers
784 views

Quaternion Differentiation

I have an application that tracks an image and estimates its position and orientation. The orientation is given by a quaternion, and it is modified by an angular velocity every frame. To predict the ...
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1answer
421 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
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1answer
330 views

how to get rotation component of quaternion form using 3d coordinates

I have a series of 3d coordinates distributed in a 3d space according to a root point. I can determine the x, y , z component using reducing the vectors. but I am not clear how to get rotation ...
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8answers
4k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
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1answer
298 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
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1answer
522 views

Quaternions and Rotations

Two of the interesting achievements in Mathematics are Classification of platonic solids, and also classification of finite groups acting on the unit sphere in $\mathbb{R}^3$, and they are very nicely ...
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0answers
219 views

Quaternions in olympiad 3d geometry

It's known that we can use complex numbers to solve some 2d problems easier than synthetic methods. But, what do you think about using complex numbers in 3d geometry? I've found extend of complex ...
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1answer
91 views

Why is the square of quaternions not half of all axes angles expressed by this quaternion?

I want to devide a rotation, which is expressed as a quaternion. So I am doing it with Quaternion^POWER, where power is lower than 0. See my question before: here If I calculate following example ...
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2answers
150 views

Why does $i$ have infinitely many conjugates in $\mathbb{H}$?

Browsing this question: Why are the solutions of polynomial equations so unconstrained over the quaternions?, the pdf linked in the comments says that the infinitely many conjugates of $i$ in ...
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1answer
5k views

Compute Angle Between Quaternions (in Matlab)

I am working on a project where I have many quaternion attitude vectors, and I want to find the 'precision' of these quaternions with respect to each-other. Without being an expert in this type of ...
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1answer
245 views

Unit elements in Hurwitz quaternions

Hurwitz quaternions are defined as: $$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$ (that is, all integer or half integer quaternions). ...
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1answer
137 views

Left and Right Vector bundles

I am reading a paper that starts talking about 'left vector bundles' and I'm having trouble figuring out what they mean. The specific setup is as follows: A quarternionic line bundle $L$ over ...
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3answers
770 views

How to get a part of a quaternion? e.g. get half of the rotation of a quaternion?

if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation? Thanks in advance!
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1answer
440 views

Splitting of quaternion algebras

A rational (definite) quaternion algebra is an algebra of the form $$ \mathcal{K} = \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha \beta $$ with $\alpha^2,\beta^2 \in ...
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0answers
462 views

How to convert Yaw, Pitch, Roll and Acceleration value to cartesian system?

I am having readings of Yaw, pitch, Roll, Rotation matrix, Quaternion and Acceleration. These reading are taken with frequency of 20 (per second). They are collected from the mobile device which is ...
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2answers
161 views

Why is any proper division subring of $\mathbb{H}$ contained in the center $Z(\mathbb{H})$?

Here is an idea I've been working on for self study. Suppose $S$ is a division subring of $\mathbb{H}$ (the quaternions, viewed as a subring of $M_2(\mathbb{C})$), which is stabilized by the maps ...
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1answer
66 views

Why do $\mathbb{C}$ and $\mathbb{H}$ generate all of $M_2(\mathbb{C})$?

For this question, I'm identifying the quaternions $\mathbb{H}$ as a subring of $M_2(\mathbb{C})$, so I view them as the set of matrices of form $$ \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} ...
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1answer
420 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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1answer
219 views

Mapping from a rotational quaternion to a single angle

I have given an attitude quaternion that describes the rotation of a device. This roation is relative to the magnetic north, so the quaternion includes implicit information about the heading of the ...
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1answer
659 views

Adding angular velocity values

I'm creating a game in which the user directly controls the tilt of a platform. Since most of the time we will be balancing and controlling a ball, this means large angles and full rotational ...
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2answers
1k views

associativity of quaternion multiplication

I understand that quaternion multiplication is non-commutative, but what association does it have. putting into context when we have the statement when given general numbers, and algebra such as: ...
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1answer
555 views

converting quaternion to vector3

I am considering taking a given rotation quaternion $Q$, about a given point $P1$, and obtaining a vector3 $V$. my thought pattern is thus: given a point $P1$, and a quaternion $Q$ obtain another ...
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0answers
70 views

A question about hypercomplex numbers: quaternions, octonions etc [duplicate]

Possible Duplicate: Why is 8 so special? First of all let me state that I am not a mathematician but I work in computer science and engineering. I was reading about hypercomplex numbers, ...
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1answer
225 views

Euler angles, quaternions and hyperspace

In three-dimensional space it's possible to define rotations using the Euler angles $(\Psi,\Theta,\Phi)$ or quaternions $(i,j,k)$ If we have a hyperspace with more than three coordinates, is it still ...
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1answer
969 views

Uniform Random Quaternion In a restricted angle range

I'm trying to sample uniform random rotations. I'd like the rotations to be restricted in a range [-θ,θ]. I found a method by K. Shoemake which can be summarized as: ...
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4answers
2k views

Quaternions and spatial translations

From my understanding, in spatial applications (3D rendering, games and similar applications) quaternions can only be used to describe rotations/orientations and not translations (like a ...
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2answers
199 views

Why is $\mathbb{H}\otimes\mathbb{H}\cong\text{End}_\mathbb{R}\mathbb{H}$?

When I first learned of the quaternion algebra $\mathbb{H}$, the most concrete way to get a grip on the ring of its endomorphisms $\operatorname{End}_\mathbb{R}(\mathbb{H})$ was to view them as ...
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5answers
2k views

Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
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1answer
136 views

strange matrix -> quaternion conversion problem

I find two different rotation matrices are mapped to a single quaternion. $$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ 0 & ...
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2answers
376 views

Does quaternion multiplication relate to Minkowski space?

A quaternion notated as $a+bi+cj+dk$ can also be written in terms of a scalar and a vector $(a,v)$, where $v$ is the three-vector $(b,c,d)$. In this notation, the real part of the product $(p,q)(r,s)$ ...
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2answers
916 views

Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset ...
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1answer
198 views

Quaternions vs Axis + angle

I have been trying to find the difference between the two but to no luck minus this: The primary diff erence between the two representations is that a quaternion’s axis of rotation is scaled by ...
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3answers
411 views

Matrix Representation of Octonions

Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following $$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ...
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0answers
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Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$. We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and ...
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1answer
1k views

Understanding the value of inner product of two quaternions in Slerp().

I'm reading pbrt and trying to better understanding the return value of Dot(). The Dot() function takes two quaternions and returns their inner product. Also note, internally, when it comes to the ...
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1answer
6k views

Quaternion distance

I am using quaternions to represent orientation as a rotational offset from a global coordinate frame. Is it correct in thinking that quaternion distance gives a metric that defines the closeness of ...
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3answers
2k views

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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1answer
294 views

Conjugation of Quaternions as Rotations in $\mathbb{R}^3$

In a lot of published texts, when I read about the conjugacy class of pure quaternions being rotations, they assume that the norm of the non-pure quaternion is 1. Is there a reason that this is ...
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2answers
252 views

what is the tensor product $\mathbb{H\otimes_{R}H}$

I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it. I have built the algebra and played with it for a ...