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

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5
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1answer
342 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 ...
1
vote
1answer
197 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 ...
2
votes
1answer
489 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 ...
2
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2answers
892 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: ...
1
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1answer
456 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 ...
1
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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, ...
0
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1answer
201 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 ...
0
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1answer
706 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
1k 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 ...
9
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2answers
185 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 ...
46
votes
5answers
1k 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 ...
0
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1answer
131 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 & ...
2
votes
2answers
285 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 $(p,q)(r,s)$ ...
9
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1answer
611 views

Why are properties lost in the 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 ...
0
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1answer
183 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 ...
2
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3answers
325 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
74 views

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 ...
1
vote
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 ...
7
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1answer
3k 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 ...
21
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2answers
1k 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 ...
0
votes
1answer
262 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 ...
3
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2answers
210 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 ...
3
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1answer
400 views

Why is quaternion algebra 4d and not 3d?

Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D. $ i^2 = j^2 = k^2 = ijk = -1 $ Using $1, i, j,$ and $k$ as the base (where ...
0
votes
2answers
223 views

Isn't it cheating to consider an $ \mathbb{R}^3 $ vector as a “pure quaternion”?

In Jack Kuipers' book he says (p 114): How can a quaternion, which lives in $\mathbb{R}^4$ operate on a vector, which lives in $\mathbb{R}^3$ ? His answer: A vector v $\in$ $\mathbb{R}^3$ can ...
4
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2answers
283 views

Square roots of $-1$ in quaternion ring

In this Wikipedia page it is said that the square roots of -1 in the quaternion ring are the elements of the imaginary sphere. I don't understand why this is so. I don't understand the system that's ...
1
vote
1answer
138 views

Decomposing a quarternion into unit axes rotations

I'm using unit quaternions to represent rotations. Given rotation angles X, Y and Z, I can construct 3 quaternions that rotate around each unit axis. Qx = ( sin(X/2), cos(X/2), 0, 0 ) Qy = ( ...
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0answers
162 views

Handedness Convention for Pauli Matrices/Quaternions as rotations in 3-space

Its a fairly standard result that one can represent rotations in 3-space with unit quaternions (or pauli matrices which can be mapped to the unit-quaternions). Working through a related quantum ...
5
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1answer
1k views

Averaging quaternions

Given multiple quaternions representing orientations, and I want to average them. Each one has a different weight, and they all sum up to one. How can I get the average of them? Simple multiplication ...
4
votes
1answer
526 views

Quaternion for an object that to point in a direction

Given two vectors, a direction and an up: how do I construct a quaternion so that when a coordinate system is transformed by it, it's X-axis points in the original direction vector?
2
votes
0answers
305 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 ...
0
votes
1answer
657 views

Quaternion between 2 3D planes

I have 2 vectors, U1 and V1 (from origin) in 3D space, together forming a plane P1. The vectors then both changes to U2 and V2 (still from origin) forming a new plane P2. Is there there a way to ...
5
votes
1answer
757 views

Using quaternions instead of 4x4 matrices for transformations

I'm interested in implementing a clean solution providing an alternative to 4x4 matrices for 3D transformation. Quaternions provide the equivalent of rotation, but no translation. Therefore, in ...
2
votes
1answer
396 views

quaternion representation of the rotation of a sphere into plane displacement

I do have a sphere of known radius which does have a coordinate frame rigidly attached to it. Let's call the coordinate frame attached to the sphere XYZs. The sphere can be rotated and displaced ...
3
votes
2answers
674 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 ...
4
votes
2answers
248 views

Quaternions as roots

So, I StumbledUpon this really cool site and the last picture looked almost as if it had 3D structure. This reminded me of another website where I saw pictures of the order-8 Mandelbulb. I got to ...
2
votes
1answer
280 views

Cohomology rings of (some) sphere bundles over spheres

Recall that 2-dimensional complex vector bundles over $S^4$ are classified by $\pi_4(BU(2))=\pi_4(BU)=\mathbb Z$. For any integer $\lambda$ one can consider projectivisation of the corresponding ...
18
votes
5answers
1k views

Can Euler's identity be extended to quaternions?

Euler's identity is $e^{i \pi} + 1 = 0$, a special case of the Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, where $\theta = \pi$ (half-turn of the unit circle). It is commonly described ...
7
votes
2answers
16k views

How do you rotate a vector by a unit quaternion?

Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
1
vote
1answer
3k views

How do I apply an angular velocity vector[3] to a unit quaternion orientation?

I have an angular velocity vector[3] in three dimensions and a unit quaternion (magnitude of 1) representing an orientation in three dimensions. I need to apply the angular velocity to the quaternion ...
3
votes
1answer
1k views

Possible to calculate Yaw,Pitch,Roll from Quaternion without using tangent?

I'm currently working on a project that involves using the Yaw, Pitch and Roll from a given Quaternion to calculate an objects orientation and acceleration. I've searched a lot about how to obtain ...
5
votes
1answer
3k views

Rotate Quaternion A by 180 degrees

Suppose you have an arbitrary quaternion - call it A - how do you rotate it by 180 degrees? Is there a way to do this without convert to angle-axis representation, i.e., keep it within the ...
0
votes
0answers
2k views

Compute Altitude and Azimuth using either Quaternions or Rotation Matrix or Roll, Pitch and Yaw component

I am struck with a mathematical problem. I want to convert the iPhone device's attitude information which is available in one of the following forms: Quaternion Rotation Matrix Roll, Pitch and Yaw ...
2
votes
1answer
243 views

Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...
3
votes
1answer
658 views

Interpolating Rotation Quaternions

Suppose I've got two quaternions that each represent an angle. I need to interpolate between these two angles (from 0% to one side to 100% to another side). Since I work a lot with complex numbers, ...
0
votes
2answers
188 views

Is the multiplication result of two non-equal unit-length quaternions perpendicular to both operands?

The question: Is the multiplication result of two unit-length quaternions "perpendicular" to both operands? "Perpendicular" means that dot product between either operand and multiplication result will ...
2
votes
1answer
64 views

Is there a specific name and/or representation for the group of distinguishable orientations of a given shape?

Consider a 3-dimensional body $B$. The space of all orientations of $B$ in 3-space is the rotation group $SO(3)$, and is often represented in computer science applications as the quaternions with $q$ ...
2
votes
2answers
116 views

$\bar{z}$ and rotation by quaternion multiplication

Does there exist a quaternion $q$ on the unit sphere such that, given the vanilla complex plane $\mathbb{C}$, $q\mathbb{C}q^{-1} = \bar{\mathbb{C}}$? Motivation: ordinarily, the plane is rotated by ...
13
votes
1answer
266 views

Quaternionic veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow\mathbb{C}P^3$ is ...
1
vote
2answers
223 views

For quaternions, is the natural log the inverse of the exponential function?

That is to ask, is $e^{\ln(q_0)}$ = $\ln(e^{q_0})$?
1
vote
2answers
449 views

Problem with my camera rotation

I am using quaternions to rotate my camera. I only rotating the camera around it's x and y axises. But for some reason, after some rotation, y axis also gets rotated(looks like around z axis). I ...