# Tagged Questions

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

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My task is: "Describe rotation $S \circ R$ by axis and angle, where $R$ is rotation around $(0,1,1)$ by 90 degrees, and $S$ is rotation around $(1,-1,0)$ by 90 degrees." I should use quaternion ...
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### Extracting the Axis a Quaternion is rotating around from the Quaternion itself Directly

Quaternion has components X, Y, Z, and W. If you created a Quaternion with input being a 3D Vector representing the axis (X,Y,Z) and a floating point number representing the amount to rotate around ...
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### Quaternion - Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). And I am trying to calculate the angle of rotation around all the three axes. I have tried may methods but not getting the ...
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### multiplication of quaternions is like complex numbers multiplication?

Suppose $p = z + j w$ where $z = x_0 + i x_1$ and $w = x_2+ix^3$. Let $q = \alpha + j \beta$ where $\alpha = y_1 + i y_2$ and $\beta = y_2 + i y_3$. How can we multiply $p$ and $q$. Is is just like ...
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### Rotation matrix between two similar cuboids using their upper sides ( and the planes defined by these sides)

I have two different images and with them an estimation of two planes ( defined in the same system). I would like to get the rotation matrix, quaternion or euler angles of a surface within this ...
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### calculating the orientation of an object

If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to ...
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### What is the $\sqrt{-1}$ when working in a quaternion space?

I dont think I really need to elaborate, do I? If you know what quaternions are then you know there are several imaginary-value options to choose from, or axes, along which the $\sqrt{-1}$ may exist. ...
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### Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
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### How to rotate in quaternions but for 2d version for arbitrary angle?

I am trying to understand the idea behind rotating in quaternions, but first I want to understand the math for 2d rotation. I saw some youtube videos, and I know that for 2D, a point in 2D can be ...
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### Angular velocity computation

Say I have two different unit quaternion $Q1$ and $Q2$ representing two different orientations in 3D space. How can I compute the angular velocity $\omega$ that would produce a rotation from $Q1$ to ...
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### Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
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### Matrix Algebra over Algebraically Closed Field

In Maclachlan and Reid's The Arithmetic of Hyperbolic 3-Manifolds, when proving that quaternion algebras are simple, they make use of the fact that $M_2(K)$, where $K$ is an algebraically closed ...
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### Is there a relationship between Rotors and the Rodrigues' rotation formula

I am trying to understand quaternion in general, and it seems like the path to making sense of how they actually work is to first understand rotors and other techniques related to rotations. By ...
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### Rotors/Quaternions: double reflection question

I am trying to learn/understand quaternion. I found this reference (among many others): http://www.geometricalgebra.net/quaternions.html It states (see attached screenshot of that page), that to ...
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### Why are quaternions (over the field of complex numbers) two-rowed matrix algebras?

I've enjoyed D.E. Littlewood's Skeleton Key to Mathematics up to chapter 8. After chapter 8, increasingly more statements are left without proof and I'm lost. He writes [...]it can be shown ...
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### Map from unit quaternions to SO(3)?

On the wikipedia page for "Rotation Group SO(3)" I read that there is a 2:1 surjection from the unit quaternions, $q=w+xi+yj+zk$, to the rotatation matrix Q= \left( \begin{array}{ccc} 1-2y^2-2z^2 ...
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### What is the perspective projection of a 3d point relative to a quarternion encoded camera?

I'm representing a camera on the cartesian space as a tuple of a 3d point (position) and a quarternion (rotation). I get the front, right and up vectors of the camera by applying the quaternion to the ...
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### rotate geometry along curve velocity without roll

I am a programmer and I'm writing a script that turns any 3D function into a 3d tube (discrete geometry). In this example I have a bezier curve f that loops and a set of vertex offsets V that ...
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### How do Quaternions return the Rodriguez formula for rotations?

While trying to work out the general formula for quaternion rotations, I found myself having difficulty getting the result to be the same as the Rodriguez formula as is suggested by multiple works: ...
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### For quaternions, is the natural log the inverse of the exponential function?

That is to ask, is $e^{\ln(q_0)}$ = $\ln(e^{q_0})$?