# Questions tagged [quaternions]

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

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### Determining attitude from a single set of vectors.

I'm given a large list of absolute points in 3D space, as well as vectors to a subset of these points from a single unknown location, collected from something like a star tracker. I'm tasked with ...
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### Unit vectors and quaternions

This might be a dumb question but recently I came to know about the quaternion number system. I can't stop wondering, "Are they related to unit vectors in any way? They have similar notations and ...
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### Connectedness of non-zero quaternions

I am trying to prove that set of non-zero quaternions is path connected. It is $S^3$, but I am not using path connectedness of higher dimensional spheres (except $S^1$ and $S^2$). I am using the book ...
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### How to get the left and right rotation quaternions between two $\mathbb{R}^4$ unit vectors?

This question is somewhat similar to this, but in higher dimension. I have two non-collinear unit vectors in $\mathbb{R}^4$, $\bf a$ and $\bf b$. I know from Wikipedia that there is at least one pair ...
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### Why does no three-scalar parameterization of 3d orientation exist that doesn't contain singularities?

I'm reading a magazine article on 3d orientation and want to know what the mathematical issue is and where to read about it, preferably a theorem title that I can google or a topic and textbook: It’...
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### Expressing common vector operations in terms of quaternions

Since everything we do with vectors could also in theory be expressed using quaternions (or octonions & higher-order -nions), is it possible to have common vector operations like the cross product ...
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### First order, non-linear ODE

I am solving the following equation: $$f(x)^2+\frac{f(x)^2}{f'(x)^2}=h(x)$$ for a known function $h(x)\geq 0, x\in \mathbb{R}$. I'm asking for help with: Identifying this ODE in terms of well-...
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### Find approximating roots to a quaternion polynomial with a naive algorithm

I am trying to devise a Niave Algorithm to find the roots to Quaternion Polynomials. I thought I had found a solution in the article Newton Method in the Context of Quaternion Analysis, as a Newton ...
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### Quaternion operation to cancel out roll angle

I'm making experiments with quaternions by using an IMU (BNO055), the sensor can give me orientation as quaternion and euler angles by itself. I'm also practicing with quaternions. My problem is, say ...
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### Are there drawbacks replacing quaternions by Conformal Geometric Algebra in an implementation

I am working on a 3D IK engine and stumbled across Conformal Geometric Algebra. I initially did find it a bit confusing but then again quaternions still occasionally perplex me. I am strongly ...
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### Quaternion PID control

I'm not quite sure how to apply a Proportional-Integral-Derivative control to a quaternion representing orientation. There are a few things I don't understand. Given the PID control formula of: <...
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### Calculating body frame angles (Euler angles) between reference frames using quaternion rotation

Problem: Convert Euler angle orientation relative to one reference frame to a new set of Euler angles relative to a new reference frame using quaternion rotation. Example: Consider a given Yaw, Pitch ...
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### Add a custom extra rotation to a quaternion

I have a quaternion which represents a 3D object rotation, for example: Quaternion(scalar:0.707107, vector:(0, 0, 0.707107)) I want to keep the current 3D object ...
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### What does the surface |z₁|² = |z₂|² look like?

In a quaternionic plane there are 2 axes and each point corresponds to $q \in \mathbb C^2$. Now, $|z_1|^2 = |z_2|^2$ should define a surface that divides a 3-sphere $|z_1|^2 + |z_2|^2 = 1$ into two ...
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### What are “hermitian matrices” and “unitary matrices” called in the context of quaternions?

The real numbers have a conjugation operation $\overline{a} = a$, and we can define the conjugate-transpose operation $A \mapsto A^\dagger$ on matrices. (Here this is just the transpose, because the ...
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### Quaternionic quantum channels.

What is the proper definition of quaternionic quantum channel and do we have the choi kraus representation theorem for quaternionic case as well?
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I was reading "Physics from Symmetry" by Jakob Schwichtenberg recently. And in part 3 of chapter 3, he goes over quaternions and I found the following statement: The set of unit quaternions $q = a\... 1answer 28 views ### Understanding the Quaternion rotation identity proof. I am doing a project about quaternions and their rotation. I am trying to get the proof of quaternion rotation identity by Wikipedia: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation to ... 0answers 8 views ### To change a quaternion relative to its current state around an axis Say I have an object with a pose define with the quaternion [0,1,0,0], where the first three values are the vector part. How should I change this quaternion to rotate the pose 10 degree around z axis ... 1answer 31 views ### Quaternion rotationmatrix I'm a last year student at a high school in Denmark, and I'm trying to understand how quaternions rotations can be described through a rotationmatrix. As of now, I quite don't understand how I go from ... 1answer 63 views ### Show that$\mathbb{H}$is a division ring Let$\mathbb{H} \subset M_2(\mathbb{C})$be the set of matrices of the form: $$A = \begin{pmatrix}z & -\bar{w}\\w & \bar{z}\end{pmatrix} \text{ where } z,w \in \mathbb{C}$$$A^{-1} = \begin{...
I got the following problem: I want to convert the normal vector of a person into an 2D orientation represented as Quaternion. The normal vector is given only in the x-y plane : \begin{pmatrix}x &...