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Questions tagged [rotations]

This tag is for questions about *rotations*: a type of rigid motion in a space.

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Decompose 3D rotation matrix into rotation around x, y and z-axis

I have a rotation matrix R, that produces an arbitrary rotation in a 3D space. I would like to decompose it into 3 rotation matrices Rx, Ry and Rz so I can use and apply only xy in plane rotation (...
Ricardo Domingos Ferreira's user avatar
7 votes
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305 views

Solid harmonic addition theorem in higher dimensions?

The solid harmonics are solutions to Laplace's equation in spherical coordinates. The regular and irregular solid harmonics, obtained by rescaling spherical harmonics, are respectively $$R_l^m(\textbf{...
John's user avatar
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289 views

Solving Euler's equation for rigid-body rotational velocity in 4D

A rigid object, with no torques applied to it, rotates with constant angular momentum. But its angular velocity $\omega$ is not constant in general; it follows the differential equations $$\frac{d\...
mr_e_man's user avatar
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7 votes
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Rotating an $n$-dimensional hyperplane

Let $\mathcal{H}: \mathbf{x}^T\mathbf{w}+b=0$ be a hyperplane in the $n$-dimensional Euclidean space of column vectors. Is there a way of "rotating" the above hyperplane such that it coincides with ...
nullgeppetto's user avatar
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6 votes
1 answer
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Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
user153816's user avatar
5 votes
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79 views

Reflections of a point about n lines returns point to its original position

Here's a very interesting problem that I made up with a friend this morning: For which even $n$ does there exist a permutation $\pi$ of $\{1,2,\cdots,n\}$ such that when we reflect any point $P$ in ...
TheBestMagician's user avatar
5 votes
0 answers
69 views

Path of the sun across the sky in a 4D world

Someone asked a question on worldbuilding about navigating by the stars on a 4D planet. In thinking about it I came up with a question that seems appropriate to ask here, as it's purely a maths ...
N. Virgo's user avatar
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motion of a rigid cube

A rigid cube is in motion. At the time depicted in the figure the face $ABCD$ is vertical, the velocity of vertex $A$ is vertical down with value $v$, the velocity of vertex $C$ is vertical up with ...
JennyToy's user avatar
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0 answers
212 views

Euler Angles on $\mathbb{R}P^3$?

I am curious if anyone can point me to a reference where the Euler angle coordinates are visualized as a parametrization of $\mathbb{R}P^3$. Bonus points if there are visualization aids for the ...
Evan's user avatar
  • 371
5 votes
1 answer
363 views

Matrix exponential of the sum of two skew-symmetric matrices

This is my first message in this site. I'm a mechanical engineer with, amongst others, an interest in inertial navigation. I'm currently reading the book "Principles of GNSS, Inertial and Multisensor ...
Guillermo Benito's user avatar
5 votes
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What does the rotation group of $\mathbb{\bar{Q}}^n$ look like?

There's a structural difference between the rotation groups of $\mathbb{Q}^n$ and $\mathbb{R}^n$; in some abstract sense the former is 'small' (discrete?) while the latter is 'large'. I suspect that ...
Steven Stadnicki's user avatar
5 votes
1 answer
3k views

Standard Basis of $SU(2)$--where does the 1/2 come from?

The most common matrix representation of $SU(2)$ is given by $$ \begin{pmatrix} a & b\\ b^* & -a^*\\ \end{pmatrix} $$ where $a,b\in\mathbb{C}$. If we denote real components by the subscript $...
JMJ's user avatar
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4 votes
1 answer
80 views

rotationally invariant matrix function

Consider a function $f:\mathbb{R}^{N \times M} \to \mathbb{R}^{N \times M}$, that takes a matrix $\mathbf{A} \in \mathbb{R}^{N \times M}$ as input and the output is a matrix of the same size. Suppose ...
Rostam22's user avatar
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Which rotation groups of compact subsets of $\mathbb{R}^n$ are symmetry groups of compact subsets of $\mathbb{R}^n$?

Definition. Let $G < SO(n)$ be the rotation group of some compact subset of $\mathbb{R}^n$ We call $G$ chiral if there exists a compact set $K$ such that its symmetry group equals $G$. Question: ...
Arshak Aivazian's user avatar
4 votes
0 answers
311 views

Distribution of a point on a sphere after a random rotation is applied twice

Pick a random rotation uniformly in SO(3), where the uniform distribution is the unique distribution that's invariant by the action of any group element. Apply this rotation twice to the north pole of ...
Arthur B.'s user avatar
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Finding the parameters of curves of rotations in rotation space

I asked this a while ago: For more background information.... Not required for this question... 2020/07/24 - changes at end I have a vector, it contains 3 curvatures. A curvature is a change in ...
J Decker's user avatar
  • 111
4 votes
1 answer
60 views

A question about rotation of a plane

Suppose for any subspace $F$ of $\mathbb{R}^d$(with the usual Euclidean norm), $\pi_F$ denote the orthogonal projection onto $F$. Let $R$ be a rotation of $F$ and $F^\prime = RF$. Prove that $\forall ...
Sudipta Roy's user avatar
4 votes
0 answers
419 views

Rotation invariants for higher degree homogeneous polynomials (like Tr$(P^m)$ for degree 2)?

Treating rotation in $\mathbb{R}^n$ as $x\to Ox$ for orthogonal $O^T O=O O^T=1$, we can easily get complete sets of independent rotation invariants for degree 1 and 2 homogeneous polynomials: Degree ...
Jarek Duda's user avatar
4 votes
0 answers
68 views

Why don't more celestial bodies exhibit higher-order rotations?

It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have ...
404UserNotFound's user avatar
4 votes
4 answers
3k views

Angle of rotation based on direction cosines

I have a question which is bothering me for days! Suppose that we have a fixed frame $XYZ$ and a moving frame $xyz$ in 3D. The moving frame is orthonormal and is defined based on the fixed one using 9 ...
Joe Hofstrand's user avatar
4 votes
0 answers
133 views

Free groups of rotations of the sphere

Is the following conjecture true: If $G$ is a group of rotations of the sphere and $G$ contains two noncommuting rotations of infinite order, then $G$ has a free subgroup of rank $2$. By the Tits ...
stan wagon's user avatar
  • 1,663
4 votes
0 answers
266 views

Check if a point is inside a rotated 2D NACA 0012 airfoil

I've already checked the rotated rectangle problem but this is (I think!) a little more complicated. I have a CFD calculation of a 2D NACA 0012 airfoil and I need to test if a point is inside the ...
André Almeida's user avatar
4 votes
0 answers
867 views

net of oblique cone,why it has a shape like this?

today i was building a right cone for my geometry homework.after building the cone, i started to think what shape the net of an oblique cone (cones with circular base which the axis does not pass ...
user2838619's user avatar
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4 votes
0 answers
735 views

Rotate a 3D Vector onto Another 3D Vector

I am trying to transform one triangle onto another triangle in 3D space (Right Triangles). My thought was I align the forward and left vectors, then translate the center of one to the other. ...
user2970916's user avatar
4 votes
0 answers
244 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
Steven Stadnicki's user avatar
4 votes
1 answer
271 views

Infinite dimensional reps of the rotation group

$\mathbf{Background:}$ The following is paraphrased from ``Representations of the rotation and Lorentz groups and their applications,'' by Gel'fand. Consider a finite-dimensional representation $T: ...
marlow's user avatar
  • 793
4 votes
1 answer
696 views

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 ...
Elto Desukane's user avatar
4 votes
0 answers
299 views

Infinite product of rotation matrices

Suppose we have a product $$\vec v=\left(\vec x^T \cdot R(\vec\varphi_1)\cdot R(\vec\varphi_2)\cdot ...\right)^T,$$ where $R(\vec\varphi_i)$ is a matrix of rotation by $3D$ angle $|\vec\varphi_i|$ ...
Ruslan's user avatar
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4 votes
0 answers
154 views

Problems sampling from a $pdf$ over $SO\left(3\right)$

I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series: $$ f\left(\omega,\theta,\phi\right)=\sum s_{\...
okj's user avatar
  • 2,509
4 votes
3 answers
1k views

Unit Quaternions on the 3-sphere, $S^3$ as orthogonal transformations.

I am reading through Andrew Hanson's "Visualizing Quaternions" and came across this passage on page 50: $q(\theta, {\bf n}) = \left( \cos\frac{\theta}{2}, {\bf n} \sin \frac{\theta}{2} \right)$ ...
Evan's user avatar
  • 371
3 votes
0 answers
47 views

Shortest paths in SO(3)

In $\mathbb{R}^3$, we have a sense of "shortest path" between a pair of points and can easily project nearby points in $\mathbb{R}^3$ onto this path. Specifically, given $a, b, c\in\mathbb{R}...
Tom's user avatar
  • 480
3 votes
0 answers
236 views

I need a rigorous explanation of Shoemake method to sample uniformly from the group of unit quaternions

I know and understand the subgroup algorithm to sample from uniform distribution on the rotation group $SO\left( 3 \right)$, following the following steps: sample $\theta_{1}$ from $\text{Unif}\left[ ...
petem's user avatar
  • 630
3 votes
0 answers
89 views

If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

As in the question title, let $A, B$ be a partition of the unit circle $S^1$, equipped with the Haar measure. Here, we do not require $A, B$ to be measurable. Also, assume neither $A$ nor $B$ is of ...
David Gao's user avatar
  • 6,805
3 votes
0 answers
92 views

Differentiable formula that computes rotations in interval arithmetic (bounding box of a family of rotated rectangles)

I want to know how to perform a rotation in 2D interval arithmetic. That amounts to computing the tightest interval containing $$ x\ \mathrm{cos}(\varphi) + y\ \mathrm{sin}(\varphi), \tag{1}$$ where $...
Will's user avatar
  • 1,800
3 votes
0 answers
312 views

What is the vorticity of a velocity field?

If $u:\mathbb{R}^3\to \mathbb{R}^3$ is a velocity field one defines the vorticity as the curl \begin{align} \omega= \text{curl}(u). \end{align} I just read that vorticity measures the rotation of the ...
homersimpson's user avatar
3 votes
0 answers
42 views

How does turtle geometry keep track of which way is up/down?

I'm having an issue understanding how turtle geometry is supposed to work. Let's say for the orientation vector O the Heading, Left, and Up unit vectors for the turtle are: So the turtle is facing ...
MFerguson's user avatar
  • 137
3 votes
0 answers
33 views

Are balanced measures isotropic?

Consider a Borel measure $\mu$ on $\mathbb R^n$ ($n\geq2$) of finite second moment, i.e. such that $$ \int_{\mathbb R^n} \|x\|^2 \,\mathrm d\mu(x) < \infty. $$ It is said balanced in the direction $...
Cryme's user avatar
  • 504
3 votes
0 answers
286 views

How best to convert Tait-Bryan angles to Euler Angles? For VR Controller project.

Again I have played with magicks beyond my understanding. I have a unique problem: I'm trying to interface three programs that no one has ever tried, in an effort to make a cheap VR controller. I am ...
TheGeek007's user avatar
3 votes
0 answers
103 views

Is any closed subgroup of $SO(n)$ a rotation group of some compact subset of $\mathbb{R}^n$?

I proved this in dimensions $n = 1, 2, 3$ from the direct classification of subgroups. For $n \geq 4$ I can't understand anything. Also, I can't find any information about this issue. Is this a solved ...
Arshak Aivazian's user avatar
3 votes
0 answers
81 views

Representations of $SO_3(\mathbb{R})$ and their characters

I am trying to understand the representations of $SO_3(\mathbb{R})$. Consider the space $P_n$ of homogeneous polynomials of degree $n$ in $(x,y,z)$. I want to understand the characters of $V_n = \ker (...
Wolker's user avatar
  • 1,077
3 votes
0 answers
48 views

Rotating a 3D shape so that it gets heated evenly by a fire

Imagine you have a shape (say, an eggplant) that you want to cook roughly evenly on a fire. How should you rotate the eggplant to accomplish this? More concretely, the surface of the eggplant (before ...
chausies's user avatar
  • 2,210
3 votes
0 answers
158 views

Rodrigues equation VS Hamilton's quaternions. Historical confusion.

I don't have a special math education and when I study quaternions I spent a long time. Along the way, without realizing, I independently derived the Rodrigues equation, because now we know about the ...
OpenglNoob's user avatar
3 votes
0 answers
116 views

Rotation matrices for N dimensional rotation about N-2 dimensional subspace

I have been looking into the generalisation of rotation about an axis in 3D. Which I have found to be is rotation of a vector in N dimensional space about an N-2 dimesnional subspace. The paper ...
Parmeet Singh EP 066's user avatar
3 votes
0 answers
50 views

Pulling Some Threads of the 2nd Order PDE Technique

I have some conceptual questions regarding a solution technique for second order linear PDEs. The example I have been considering is $u_{xx} + 2u_{xy} + u_{yy} = 0$. The technique is to use the guess ...
user10478's user avatar
  • 1,912
3 votes
0 answers
340 views

Generators of Rotation group

Let $J_x, J_y, J_z$ be the generators for rotation about the $x, y ,z$ axis. That is, $\exp[{\theta J_i}]$ is a rotation about the $i$ axis by an angle $\theta.$ Furthermore, $\exp[{ \theta_x J_x + \...
Jbag1212's user avatar
  • 1,598
3 votes
0 answers
61 views

Vector Transformation Under an Euler Angle Passive Rotation

Let there be some object in $\mathbb{R}^3$ centered at point $P$ with coordinates $(x_P,y_P,z_P)$. Its orientation is defined by Euler angles $\alpha, \beta, \gamma$ with respect to some reference ...
Researcher_Witty's user avatar
3 votes
0 answers
49 views

Find the matrix of the rotation matrix $U$

The matrix of $U$ in $\mathbb{R^3}$,with the standard inner product which is rotation of the plane $W=sp\{\alpha_1,\alpha_2\}$ about the orthogonal line $\alpha_3$ through the angle $\theta$ , where $\...
Antimony's user avatar
  • 3,442
3 votes
0 answers
252 views

Transforming a quaternion between two coordinate systems

Background I am trying to convert an orientation quaternion from a head-mounted IMU into a quaternion which axes would correspond to the pitch/yaw/roll of said head. Since I am a quaternion novice, I ...
JohanPI's user avatar
  • 131
3 votes
0 answers
393 views

Maximizing tr(AB) under a rotation matrix constraint for B? Related to von Neumann's trace inequality

Let $\mathbf{A}$ and $\mathbf{B}$ be real matrices with $\mathbf{A}\overset{\mathrm{SVD}}{=}\mathbf{U}_A \mathbf{S}_A \mathbf{V}_A^\mathrm{T}$. I want to maximize \begin{align} \max_B \mathrm{trace}(\...
user avatar
3 votes
0 answers
32 views

Is any $m$-dimensional strict vector subspeace (so $m<p$) a rotational image of the Euclidean subspace $\mathbb{R}^m$?

Let $V$ be a vector subspace of $\mathbb{R}^p$ of dimension $m$, where $p < m$. I'm checking with you if there alawys exists a rotation $A \in O(p)$ so that $A(V) = \mathbb{R}^p \times \{0\}^{p-m}$,...
Learning Math's user avatar

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