# 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 (...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $V_n = \ker (... • 1,077 3 votes 0 answers 48 views ### Rotating a 3D shape so that it gets heated evenly by a fire Imagine you have a shape (say, an eggplant) that you want to cook roughly evenly on a fire. How should you rotate the eggplant to accomplish this? More concretely, the surface of the eggplant (before ... • 2,210 3 votes 0 answers 158 views ### Rodrigues equation VS Hamilton's quaternions. Historical confusion. I don't have a special math education and when I study quaternions I spent a long time. Along the way, without realizing, I independently derived the Rodrigues equation, because now we know about the ... • 173 3 votes 0 answers 116 views ### Rotation matrices for N dimensional rotation about N-2 dimensional subspace I have been looking into the generalisation of rotation about an axis in 3D. Which I have found to be is rotation of a vector in N dimensional space about an N-2 dimesnional subspace. The paper ... 3 votes 0 answers 50 views ### Pulling Some Threads of the 2nd Order PDE Technique I have some conceptual questions regarding a solution technique for second order linear PDEs. The example I have been considering is$u_{xx} + 2u_{xy} + u_{yy} = 0$. The technique is to use the guess ... • 1,912 3 votes 0 answers 340 views ### Generators of Rotation group Let$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 + \...
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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 ...