Questions tagged [rotations]
This tag is for questions about *rotations*: a type of rigid motion in a space.
-2
votes
1answer
17 views
$\operatorname{SO}(3)$ and Tensors of higher ranks [on hold]
I am studying group theory. I know how vectors transform under $\operatorname{SO}(3)$, but I don't know how tensors of rank $2$ or greater than $2$ transform under $\operatorname{SO}(3)$. Can someone ...
1
vote
1answer
38 views
Find a Givens rotation matrix such that $y=Gx$
Assume that $x,y \in \mathbb{R}^2$ with $||x||_2=||y||_2=1$.
Find a Givens rotation matrix $G=\begin{bmatrix}c & s \\ -s & c \end{bmatrix}$ (i.e., find $c$ and $d$ with $c^2+d^2=1$) such ...
0
votes
0answers
18 views
Angle-Axis Parametrization of SO(3) Proof
Suppose we have an element $R$ of $SO(3)$. $R$ is characterized by, $R^T = R^{-1}$. There are a number of equivalent characterizations such as $R$ preserves norms or dot products.
I am looking for a ...
0
votes
1answer
26 views
Position of points in an isometric image after rotation
I'm developing a game using isometric images for the units.
Each units has weapons represented in red on the first image below.
To attach animations to the weapons, I have to be able to calculate the ...
0
votes
0answers
17 views
Composition of uncertain rotations
Here, in this tutorial: http://ethaneade.com/lie.pdf, the author gives the composition of uncertain rotations for Gaussians in $SO(3)$ (Eqn. 47, Page 7). The author doesn't give the detailed ...
0
votes
0answers
12 views
Relationship between Cayley transform and polar decomposition
I want to (as) efficiently (as possible) numerically compute the rotation $\mathrm{R}$ in the polar decomposition of a $n\times n$ matrix of the form $\mathrm{I} + \mathrm{W}$ where $\mathrm{I}$ is ...
0
votes
0answers
7 views
Reflecting/Rotating a 2 dimension manifold in a 3 dimensional space
I have a convex tilted hexagonal figure (6 corners) in 3-D space (imagine a tilted hexagon floating in a cube). I acquire this object by applying a projection $Q$ onto a cuboid. So a point of the ...
1
vote
1answer
34 views
General equation for projection of regular grid onto a line?
I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $\tan \alpha$:
I would like to derive a mathematical ...
1
vote
2answers
52 views
Exercises about rotations
$1.$Find the axis and rotation angle of $T$ such that $T(v)=w$, for $v=(2,0,2)$ and $w=(0,2,-2)$. In case there is not such rotation, explain why.
$2.$ Say if it is possible to define a rotation ...
1
vote
0answers
24 views
Why convert Quaternion to Euler Angle
I've recently played with the IMU filter in MATLAB. When using their examples, they always plot the rotations by stating something alike this:
...
0
votes
1answer
27 views
Rotation problem [duplicate]
With a 2D surface, we take $(2, 1)$ as the center point and consider a transformation with a rotation angle of $45^\circ$ so point $(3, 3)$ is transformed into point?
I'm really close to getting the ...
0
votes
0answers
17 views
Rotate and Translate Irregular Polygon to X-axis
I need to rotate and translate an irregular polygon so that a chosen edge is on the x-axis and the "inside" of the polygon is above the x-axis.
I know how to translate and rotate a polygon using a ...
0
votes
0answers
37 views
Changing rotation center
Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
2
votes
0answers
37 views
What are the details of this step in a proof of the Banach-Tarski paradox?
In this exposition of the Banach-Tarski paradox by Terry Tao, Corollary 1.4 says,
There exists a partition $S^2 = \Gamma_1 \uplus \dots \uplus \Gamma_8$ and rotation matrices $R_1, \dots, R_8 \in ...
1
vote
1answer
34 views
Least square rotation from a set of normals to a set of normals
I know how to calculate the rotation matrix given $n$ and $m$ (both norm equal to 1) and I would like to find rotation matrix $R$ such that $m = Rn$.
now assume I have $\{n_i,m_i\}_{i=1}^K$, how do I ...
0
votes
0answers
11 views
Rotation number of a certain curve
I was asked about the rotation number of a particular kind of closed immersed curve.
Closed immersed curve, $c(t)=c(t+T), \forall t.$
Moreover, $c(t+\frac{T}{7})=R(c(t))$, where R is the rotation ...
1
vote
2answers
58 views
How to generate a rotation matrix given an angular momentum matrix
In 3 dimensions, the total angular momentum (for $z$) matrix is given. It generates the rotation matrix around $z$ by $e^{-i\theta J_3/h}.$ My question is how do we actually go about doing this? I ...
0
votes
1answer
24 views
Group of rotations for tetrahedron
Find the group of all rotations (both proper and improper) of a regular tetrahedron.
I understand that the number of elements in the group of proper rotations is 12 but I don't know how to express ...
0
votes
1answer
19 views
How to create a Quaternion rotation from a forward- and up- vector? [closed]
I need the rotation Quaternion of an object, I have it's foward and up directions (as 3D vectors), so I thought it would be easy to create a Quaternion rotation from that, but I can't seem to get it ...
0
votes
0answers
40 views
3D rotation matrix with zero determinant
I have a 3D vector from the following shape:
\begin{gather}
p = \begin{bmatrix} 0 \\ 0 \\ z \end{bmatrix}
\end{gather}
I had like to find a rotation matrix that will make p(1) and p(2) small non ...
0
votes
0answers
16 views
How to fix banking-drift in 3d road/path-data
I'm working on something in Unity3D (the game engine) where I have to modify a path/road in 3d space. The path consists of a collection of points that each have a position (Vector3) and orientation (...
1
vote
1answer
19 views
Solve for the inverse of $\mathbf I - \tan(\frac{\phi}{2}) \mathbf {\hat \omega}$
Original problem comes from some notes on rotations (at the last page), which was devoted to deriving Rodrigues' rotation formula. The complete problem is to show why
$$(\mathbf I - \tan(\frac{\phi}{2}...
1
vote
1answer
43 views
How to rotate a path in 3D (computer sciences)
I'm working on something in Unity3D (the game engine) where I have to modify a path/road in 3d space. The path consists of a collection of positions and quaternion-orientations (the orientation ...
0
votes
2answers
43 views
Rotation matrices vs quaternions?
It seems we can describe every rotation in $SO(3)$ by at least one unit vector axis $u$ and angle $\theta$ pair.
Each of these pairs can also be described by a rotation matrix:
I've heard ...
-1
votes
1answer
106 views
Including rotational motion into a reaction-diffusion model
The reference below describes a system of hypothetical sub-particle units or etherons, diffusing from a region of high to low concentration using Fick’s law of diffusion. How would one introduce ...
0
votes
1answer
17 views
Coordinates of line in sphere with x,y rotation
Lets say that I have a line with one end fixed to the center of a sphere, and the other end can freely rotate. If I were to rotate the line around the x and y axes, what would the coordinates be for ...
0
votes
1answer
33 views
rotation in complex plane.
in complex plane we have a line passing through $z_1$ and $z_2$. I want to find a line making an angle $\theta$ with this line and passing through $z_1$. How do I do this?
I know i can convert it to ...
0
votes
1answer
98 views
rotate the helix using the rotation (Rz and Rx) equations
So I want to rotate the helix
$$
\begin{cases}
x=\cos(t),\\
y=t,\\
z=\sin(t),
\end{cases}
$$
so that it wraps ground a vector $(X,Y,Z)^T$. I first get the theta by
$$
\theta=\arctan\left(\frac{Z}{Y}...
1
vote
2answers
41 views
Closest 3x3 rotation matrix where all entries are in {1, 0, -1}
Let $M$ be a 3x3 right-handed rotation matrix.
I need to find a closed form solution for the right-handed rotation matrix $M'$ where its entries are lie in ${1, 0, -1}$.
An initial idea is to replace ...
23
votes
12answers
3k views
Stuck on a Geometry Problem
$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of ...
0
votes
0answers
22 views
Haar measure from axis-angle representation of $SO(3)$
My aim is to study some special representations of $SO(3)$ using characters, and to do this I need an explicit expression of the Haar measure on $SO(3)$. I've found some "versions" of the Haar measure ...
0
votes
0answers
13 views
error dynamics of rotation matrix (derivative of logrithm of ratation matrix)
In a paper, the author defined an error term as $e = \log(R^T\hat R)$, where $R, \hat R \in SO(3)$,$R$ is the true state and $\hat R$ is the estimated state. Then the author derived error dynamics by ...
1
vote
0answers
37 views
Why don't we generalize rotation by rotating parallel to a plane instead of around a point or line?
When we learn about rotation, we are thought that in 2D you can rotate objects around a point and in 3D you can rotate things around a line. If we generalize this, then rotating an object in a $n$-D ...
0
votes
0answers
14 views
Switching rotation matrix in order of multiplication
Suppose I have Givens matrices (as defined here), $G_1,...G_k$, and some arbitrary (not necessarily invertable) square matrix $X$. Let $H_1=G_1G_2... G_k$ I want to find a matrix $H_2$ such that:
$$...
1
vote
1answer
29 views
Purpose of rotation of a Function or Graph
You are able to rotate any function by an arbitrary angle around the origin using the formula,
$$y\cos\theta-x\sin\theta=f(x\cos\theta+y\sin\theta)$$You can also do similar rotations for polar graphs, ...
0
votes
1answer
73 views
Given six 3D vectors a,b,c,d,e,f, find the rotation matrix R such that Ra, Rb, Rc are respectively perp. to d,e,f (assuming there exists a solution)
(EDIT: I also welcome a solution to the special case where d, e, f are coplanar)
I think this problem can be solved iteratively, but I was wondering if there could be a closed-form solution...
...
1
vote
2answers
44 views
Interpolation between three-dimensional rotations
I have to define a continuous function $g: [0, 1] \rightarrow \mathrm{SO}(3)$ such that $g(0) = I$ and $g(1) = R$ (a given rotation).
I know we can do this kind of interpolation using quaternions ...
0
votes
2answers
18 views
Matrix transformations on x and y axis
For physics/linear algebra I'm supposed to transform the big sideways F into the small upright F using a matrix, and then do the opposite. I'm wondering if anyone can shed some light on how to do this,...
1
vote
1answer
21 views
Rotation around the axis
I wish to rotate a body that is located on $\hat z$ axis.
If I rotate the body at angle $\alpha$ around the $\hat x$ axis and then at angle $\beta$ around the $\hat y$ axis then I think I should get:
...
0
votes
2answers
45 views
Rotation of an angle that has another as the center
"With a 2 dimensional surface:
We take (2, 1) as the center point and consider
a transformation with a rotation angle of 45◦ so point (3, 3) is transformed
into point ...?"
I'm really close to ...
1
vote
3answers
47 views
Rotating a regular tetrahedron so it looks like an egyptian pyramid?
I have been able to easily create a mesh of a regular tetrahedron thanks to this answer:
However, as you can see, it looks like it's sitting on one of its edges, what I was looking for was for it to ...
0
votes
1answer
38 views
Real $2\times2$ matrices of finite order and rotational matrices
Let $M$ be a real $2\times2$ matrix
$$ M = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right) $$
Suppose $M$ has a finite order (thus there is some natural $n$ that $M^n=E$, where $E$ is ...
1
vote
0answers
47 views
Question on Partial derivatives in inverse function changing from coordinate systems
I would like to ask a question about partial derivatives in the context of Rotations of coordinate systems. Say we have a coordinate system (unprimed) and its rotated version (primed). If the ...
0
votes
0answers
40 views
Representations of $SO(3)$ in $\mathbb{C}[x,y,z]$
So my professor has given me the problem of examining the representations of $SO(3)$ in $V = \mathbb{C}[x,y,z]$. The thing is that he's now gone away for a conference trip, and I have absolutely no ...
1
vote
1answer
119 views
Fourier Transform of a Spherical Well Potential - Rotating the System
I'm attempting to take the Fourier transform of the following function:
\begin{equation}
V(\mathbf{r})
=
\begin{cases}
V_0 & r<r_0
\\
...
0
votes
4answers
83 views
Real eigenvalues of a rotation matrix
Suppose the rotation matrix
$$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$
has real eigenvalues. Then what would the value of $\theta$ be?
I know that the eigenvalues ...
2
votes
1answer
44 views
Does the Fourier transform commute with the spherical average operator?
$\newcommand{\Rcal}{\mathcal R}$For $f\colon \mathbb R^d\to \mathbb R$, write $x\in \mathbb R^d$ as $x=r\omega$, with $\omega\in\mathbb S^{d-1}$, and define
$$
R f(r):=\int_{\mathbb S^{d-1}}f(r\omega)...
0
votes
1answer
42 views
Given a unit vector $\vec{n}$, find the matrix of rotation about $\vec{n}$.
In $\mathbb{R}^3$ suppose I have an arbitrary vector $\vec{n}$. I want to find the rotation matrix about $\vec{n}$ through an angle $\theta$.
I can develop the rotation matrix in two dimensions and ...
0
votes
0answers
16 views
0
votes
1answer
41 views
Show that rotation preserves norms for three-dimensional vectors
From S.L Linear Algebra:
Let $F$ be a rotation through an angle . Show that for any vector $X$
in $\mathbb{R}^3$, we have $||X||=||F(X)||$ (i.e. $F$ preserves
norms), where $(a, b)=\sqrt{a^2+b^...
