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

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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 ...
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2answers
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 ...
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1answer
594 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?
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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
796 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 ...
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1answer
843 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 ...
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1answer
454 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 ...
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2answers
759 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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2answers
262 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 ...
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1answer
288 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 ...
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5answers
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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 ...
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2answers
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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 ...
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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 ...
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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 ...
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1answer
4k 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 ...
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0answers
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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 ...
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1answer
250 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 ...
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1answer
778 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, ...
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2answers
198 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 ...
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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 $q$ ...
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2answers
117 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 ...
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1answer
270 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 ...
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2answers
228 views

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

That is to ask, is $e^{\ln(q_0)}$ = $\ln(e^{q_0})$?
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2answers
483 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 ...
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2answers
209 views

Why u*v = cross(u,v)-dot(u,v) at quaternions?

Why is it that for quaternions, $u*v = \mathrm{cross}(u,v)-\mathrm{dot}(u,v)$? I wonder for what reason they are equal to each others.
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The geometry of $\operatorname{PSO}(4)$ and the quaternions

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
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2answers
780 views

How do quaternions represent rotations?

I wonder how $qvq^{-1}$ gives the rotated vector of $v$. Is there any easy-to-understand proof for it? I was on Wikipedia, but I could not understand the proof there because of the conversions. Why ...
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1answer
1k views

Quaternion and rotation about an origin and an arbitrary axis origin help

Greetings All Thanks to James and Chas for helping me get this far btw Chas the language I wrote it in is in matlab. I tried to respond to your post but wasn't able to do it..I guess the gremlins ...
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1answer
3k views

Axis Angle to quaternion and quaternion to Axis angle question

Axis Angle to quaternion and quaternion to Axis angle question Greetings All (matlab / octave code) Link to text file in case formatting gets messed up http://db.tt/nVv8Ivj I created two functions ...
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2answers
2k views

How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?

How would I create a rotation matrix that rotates X by a, Y by b, and Z by c? I need to formulas, unless you're using the ardor3d api's functions/methods. Matrix is set up like this ...
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1answer
143 views

Question regarding practical SLERP

We are suppose to compute the quaternion which performs 1/5 of the rotation of this quaternion: [ 0.965 (0.149 -0.149 0.149)] The answer provided is shown as ...
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3answers
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Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
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4answers
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Quaternion Division

If $q$ and $r$ are quaternions, and $p$ is a point, applying $q$ then $r$ to $p$ is: $$ (qr)p\dfrac{1}{qr} $$ What if I want to go the other way? Instead of concatenating rotations, I want to remove ...
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7answers
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How can one intuitively think about quaternions?

Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the ...
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1answer
314 views

Differences in Quaternion properties?

In Quaternion mathematics, normally $$i ^ {2} = j ^ {2} = k ^ {2} = i*j*k = {-1}$$ But I am looking at a different type of Formula where $$i ^ {j ^ k} = {-1} $$ (edit again, since somehow this ...
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0answers
407 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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2answers
2k views

the logarithm of quaternion

I'm reading 3D math primer for graphics and game development by Fletcher Dunn and Ian Parberry. On page 170, the logarithm of quaternion is defined as \begin{align} \log \mathbf q &= \log \left( ...
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3answers
893 views

On multiplying quaternion matrices

Both matrix multiplication and quaternion multiplication are non-commutative; hence the use of terms like "premultiplication" and "postmultiplication". After encountering the concept of "quaternion ...
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2answers
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Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? Here a division algebra is an associative algebra where every ...
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6answers
5k views

Real world uses of Quaternions?

I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real ...