Articles of quaternions

Quaternion – Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope) and am trying to calculate the angle of rotation around all three axes. I have tried many methods but am not getting the results as expected. Methods tried: (i) Complementary filter approach – I am able to get the angles using the […]

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 $2^n – ion$ product. I tried it out for 3 dimensional cross product (as imaginary part of quaternion product) […]

Ideal in a matrix ring

Edit As Kimball point out, in the following question for me an ideal $I$ is a full $\mathbb{Z_p}$-lattice of $M_2(\mathbb{Q}_p)$ such that $$\lbrace\alpha \in M_2(\mathbb{Q}_p)\vert \alpha I \subset I \rbrace=M_2(\mathbb{Z}_P). $$ Let $p$ be a prime number and consider the ring, formed by the elements $$ \begin{pmatrix} a & b \\ pc & d \end{pmatrix}$$ […]

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 transformation matrix does). This seems to be backed up by the fact that most 3D frameworks that use quaternions (OGRE3D for example), use quaternions together with vectors to describe an […]

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the right Ore condition when for $x,y\in R$ the right principal ideals generated by them have a non-empty intersection. I can infer from what I’ve seen in various […]

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 $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$ and $S^2\to\mathbb CP^{2n+1}\to\mathbb HP^n$. Question. Does octonionic Hopf fibration $S^7\to S^{15}\to\mathbb OP^1$ give rise to a fibration $\mathbb HP^3\to\mathbb OP^1$? (On one hand, constructing a map $\mathbb HP^3\to\mathbb OP^1$ […]

Quaternion–Spinor relationship?

I’ve known for some time about the rotation group action of the (‘pure’) quaternions on $ \mathbf{R}^3 $ by conjugation. I’ve recently encountered spinors and notice similarities in their definitions (for example, the use of half-angles for rotations). Is the relationship that this suggested in my mind a real one, and if so what is […]

Fractal derivative of complex order and beyond

Is there some precise definition of “complex (fractal) order derivative” for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would like to know if some mathematician has defined a complex order derivative valid without restrictions for all complex number z. I mean: Is a well-defined definition […]

How to use the quaternion derivative

I’m having troubles using the Quaternion derivative. So I have: q(t) = … <– my current attitude The derivative (using w(t) as body rotation rate) is: dq(t)/dt = 1/2 * q(t) * w(t) So how do I use that? When doing stepwise integration (like in a time stepped simulation), I would have expected: q(t+∆t) = […]

Mean value of the rotation angle is 126.5°

In the paper “Applications of Quaternions to Computation with Rotations” by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by integrating quaternions over the 3-sphere). How can I make sense of this result? If rotation angle around a given axis can be […]