Questions tagged [rotations]
This tag is for questions about *rotations*: a type of rigid motion in a space.
3,160
questions
1
vote
1
answer
86
views
Is There a Rotationally Invariant Order of Points on the Sphere $S^2$
Question: Let $S^2$ denote the 2-sphere embedded in $\mathbb{R}^3$. Consider the group $SO(3)$ of rotations acting on $S^2$. Is there a strict total order $<$ on the point set $P\subseteq S^2, \...
- 321
0
votes
0
answers
32
views
Point of contact between an ellipse $(5 \cos t , 3 \sin t)$ and an apparent tangent $ x \cos(R) + y \sin(R) =D$ sliding on it.
Let $E$ be an ellipse defined by $ (x^2 / 5^2) + ( y^2 / 3^2) = 1 $ or, equivalently $( 5\cos t , 3\sin t )$ with $0\leq t \leq 2 \pi$.
Let $P= ( 5 \cos R , 3 \sin R) \space 0\leq R \leq 2 \pi$ be a ...
- 1,597
0
votes
0
answers
18
views
Find the angle of a higher dimensional rotation matrix
I was trying to find the angle of an arbitrary rotation matrix, and I decided to use the formula for angle between two vectors:
$\theta=\max\limits_{\vec{x}}\left(\arccos\left(\frac{\vec{x}\cdot R\vec{...
- 201
-3
votes
0
answers
16
views
how to get desired angular velocity with desired so3 rotation matrice?
i was reading "Geometric Tracking Control of a Quadrotor UAV on SE(3)" (available at https://arxiv.org/abs/1411.2986) and i wanted to simulate it in matlab...)
the thing is our closed ...
5
votes
2
answers
133
views
How do I minimize a function where some of the variables are dependent on other variables?
Consider a camera placed in $(XC,YC,ZC)$.
The camera is looking down on a flat table $Z=0$:
Introduce a local coordinate system for the camera: $(x,y,z)$.
Initially the axes of the local coordinate ...
- 749
0
votes
0
answers
12
views
Is this equivalence between two ways of expressing the coordinates of a rotated point useful?
Using " matrix by vector" multiplication, one can establish that the coordinates of the image ($P'$) of a point $P=(a,b)$ under a rotation by $\alpha$ radians ( counterclockwise) are :
$$ \...
- 1,597
1
vote
0
answers
16
views
Build a rotation matrix to perform discrete parallel transport on a discrete curve
Consider a discrete cuve with vertices $\{...x_{i-1}, x_i, x_{i+1}, ...\}\in \mathbb{R}^{3n}$, and denote the edge $e^i=x_ix_{i+1} $ and tangent $t_i=x_{i+1} - x_i$. When perform discrete parallel ...
- 11
5
votes
1
answer
71
views
Extension of Wahba's problem: finding two unknown rotations surrounding a known rotation
Wahba's Problem seeks to find the $3\times3$ rotation matrix which minimises:
$$ J(\boldsymbol{\mathrm{R}}) = \frac{1}{2} \sum_{k}|| \boldsymbol{{w}}_k - \boldsymbol{\mathrm{R}}\boldsymbol{v}_k||^2 $$
...
- 51
0
votes
0
answers
30
views
Retrieve Roll and Pitch rotation
I need to retrieve Roll and Pitch (but not Yaw) rotation values from rotation matrix (or euler angles, or quaternion, input can be different, fortunately, it can be converted from one form to another)....
0
votes
1
answer
47
views
Rotation in cylindrical coordinates and exterior derivative
Suppose a 1-form $A$ of $\mathbb{R}^3$ is represented as $A= A_r (r,\theta,z)dr + A_\theta (r,\theta,z)d\theta + A_z(r,\theta,z)dz$ using cylindrical coordinate system $(r,\theta, z)$.
The external ...
- 25
1
vote
1
answer
44
views
Rotation matrix and orthogonal matrix
If I am given an orthogonal matrix $A$ and I right multiply by $W $ to get $B=AW$ where $W$ is a diagonal matrix then essentially, I am scaling the columns of $A$ by the diagonals of $W$, preserving ...
- 33
1
vote
2
answers
26
views
How to recover the cartesian equation of a rotated parabola from its parametric equation obtained using linear algebra ( rotation matrix)?
Every point of the $y = x^2$ parabola can be seen as the endpoint of a position vector of the form $< t, t^2>$.
So, rotating this curve counterclockwise can be assimilated to applying a linear ...
- 1,597
1
vote
0
answers
38
views
Validity of this proof for rotation matrices being orthogonal
We were studying vector transformations using matrices and rotation matrices in our physics class and the professor's proof for some rotation matrix $R$ being orthogonal was to take a column vector $A$...
- 1,539
1
vote
2
answers
120
views
How to swap the roll and pitch of a quaternion
I'm writing code that receives quaternion values that are used to rotate a $3$D model. To display the model with the correct orientation, though, I'll need to swap the roll and pitch (rotation about ...
- 119
1
vote
1
answer
66
views
Rotations in D dimensions [closed]
So, if we are in $\Bbb R^{n}$ there are $n(n-1)/2$ rotations. Which is to say:
Given $1$ of the $n$ axis, we can rotate it onto $n-1$ other axis.
We then divide by $2$ to not overcount.
That's OK, ...
- 21
0
votes
1
answer
43
views
The area of closed curve made by sine functions that are rotated on some angle $\theta$ [closed]
I constructed interesting figure and want to know its function on Cartesian plane and the area enclosed by this figure.
Suppose you have the sine function (explicitly, $m \cdot \sin(kx)$) from 0 to $\...
- 1
0
votes
1
answer
50
views
How to see what rotation an orthogonal matrix represents? [closed]
As the output to some calculation I get an orthogonal 2-D matrix, which I know must represent some 2-D rotation (and possibly a reflection). It looks like
\begin{bmatrix}
f & g \newline
1 & 1
\...
- 323
1
vote
2
answers
46
views
Constructing transformation of an extended spherical movement
I have a frame of reference positioned on a surface of a sphere with the $z$-axis always pointing towards the center of this sphere, which is at distance $d$.
Now given:
spherical angle $\phi$
...
- 11
0
votes
0
answers
33
views
Matrix Vector multiplication of Gaussian Matrix and Gaussian Vector
I am curious about how $y$ is distributed if:
$$y=Ax,$$
with Gaussian Matrix $A \in \mathbb{R}^{m \times n}$, every entry of the matrix $e_{ij} \sim N(0, 1)$ and Gaussian Vector $x \in R^n, \mathbb E(...
0
votes
0
answers
30
views
Convert angle to value and value to angle
I have an "arc" react component and I'm struggling with the math to convert the angle (in degrees) to a value, and the value to an angle.
This arc has a "handle" which represents a ...
- 145
0
votes
0
answers
25
views
Relation between quaternions, Pauli vectors and regular vectors
I'm learning about rotations in 3D space. I've come across Pauli vectors and quaternions. Now as I understand, one can associate to a regular vector $\vec{v} = (v_x, v_y, v_z)$ a Pauli vector and a ...
- 173
3
votes
2
answers
62
views
Geometry question: equidistant center point of 45-45-90 triangle
Okay, I've been struggling on this for a couple days now. Given a 45-45-90 triangle with legs of length n, extend three perpendicular rays, one from each segment, such that they intersect within the ...
- 143
2
votes
0
answers
39
views
How to generate evenly spaced rotations in SO(3) and/or a regular grid over $\mathbb{S}^3$ (to be used to divide the space of unitary quaternions)?
I am trying to split the space of $\mathrm{SO}(3)$ into spaces $(S_i)_{i\in\{1, \cdots, n\}}$ of rotations providing a regular paving of $\mathrm{SO}(3)$.
At least I would need that $\bigcup_{i=1}^n ...
- 21
1
vote
0
answers
29
views
Sigma matrices and quaternion correspondence
I'm trying to learn the mathematics behind spinors and I've come across sigma matrices and quaternions, both used to describe 3D rotations. Now it is possible to notice that any squared pair of sigma ...
- 173
2
votes
3
answers
242
views
Solving for the rotor in quaternion rotation
If we have 2 vectors $v_1,v_2$ which have been rotated into $v'_1,v'_2$ by the following operations:
$v'_1 = e^{θ\hat{n}/2}v_1e^{-θ\hat{n}/2}$
$v'_2 = e^{θ\hat{n}/2}v_2e^{-θ\hat{n}/2}$
Where $\hat{n}$ ...
3
votes
0
answers
102
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[ ...
- 620
1
vote
1
answer
33
views
Quaternion kinematics transition matrix closed form
For rigid body rotation discrete time kinematics equation is as follows
$R_{k+1} = T_r \cdot R_k$
where:
$R$ is a rotation matrix
$T_r = e^{-[a\times]}$ is a transition matrix
$[a\times]$ means a skew ...
- 143
0
votes
1
answer
55
views
Rotation of an object in 3D
I want to rotate a 3D object using a python code. After doing some search, I was able to come up with the following initial steps:
Input the Euler axis,
input the Euler angle of rotation,
Normalize ...
- 1
0
votes
0
answers
23
views
Tifr gs 2021: symmetry of coloring chessboard under rotation [duplicate]
Let $\mathcal{C}$ denote the set of colorings of an $8\times 8$ chessboard, where each square is coloredeither black or white. Let $\sim$ denote the equivalence relation on $\mathcal{C}$ defined as ...
user1161494
1
vote
1
answer
57
views
I need an alternative 3d rotation/translation matrix for a pygame visualization.
Rotation around arbitrary axis
...
- 113
0
votes
0
answers
13
views
Relationships in formula of rotation of a plane about a point $(a, b)$ and angle $\theta$
Let $(a, b)$ be a point in $\mathbb{R}^2$ and $\theta$ be an angle of rotation. Using translation and roation matricies, namely
$$\begin{pmatrix}
1 & 0 & a \\
0 & 1 & b \\
0 & 0 &...
2
votes
1
answer
42
views
Combining rotations about different coordinate systems
I'm working on what I imagine is a common problem, but am struggling to find high quality resources. It is easiest to describe with an example:
Let's say I have a series of frames and coordinate ...
- 21
0
votes
0
answers
34
views
What convex polygons admit 60 degree rotational symmetry?
I was trying a problem from a contest,i.e rmm that involved these kind of polygons and I tried to find all polygons that admit 60 degree rotational symmetry. I believe intuitive that it has to do with ...
2
votes
0
answers
31
views
How to calculate angle of rotation given a vector and a point
sorry if the title seems vague. English is not my first language and I dont know what to search to find the answer to my problem.
I will preface this by explaining the purpose of this question. I have ...
0
votes
0
answers
28
views
Describe a rotating cube around its axis as a piecewise function
Consider a cube filled with random particles. The cube is rotated around the z-axis through its center, with the rotation being proportional to the height within the cube. At the bottom of the cube, ...
- 101
1
vote
1
answer
75
views
A single rotation to reach the final pose of a frame
The following question is from Kinematic Analysis of Robot Manipulators by Carl D. Crane, III, Joseph Duffy:
"The transformation that relates the A and B coordinate systems is given as
That is ...
- 147
0
votes
0
answers
18
views
Vector algebra of points in tangent space of SO(3) plane.
I know that we can perform algebric operations in the so(3) plane. Let us take $R_1,R_2$ and $R_3$ as rotation matrices in SO(3) plane. ^v: vector representation of skew-symmetric matrix.
Vector from $...
- 1
28
votes
9
answers
4k
views
Why do complex numbers lend themselves to rotation?
In the introductory complex analysis course I am taking, nearly every theorem relates to rotation and argument. Why do complex numbers love doing this so much?
I can understand why these theorems work;...
- 449
0
votes
1
answer
105
views
Replacing a double rotation by a single rotation?
I had a test tonight and there was one problem that I couldn't even think about a solution. Problem was this, as much I can remember:
You are doing two rotations. First one from $y$-axis to $z$-axis ...
- 851
0
votes
0
answers
17
views
Formula for Transformation of Polynomial Coefficients under Rotation
I have a function represented in a basis of 2D Legendre polynomials,
$$
f(x,y) = \sum_{n=0}^N c_n P_n(x,y)
$$
where $P_n(x,y)$ is a 2D Legendre polynomial given by
$$
P_n(x,y) = P_l(x)P_m(y)
$$
where $...
- 266
0
votes
1
answer
66
views
Why does $\boldsymbol{\nabla} \times \textbf{E}=\textbf{0}$ imply $\boldsymbol{E_2}^{\parallel}=\boldsymbol{E_1}^{\parallel}$?
I am currently studying 'Introduction to Electromagnetism' by David Griffiths, and I was reading about the electric displacement $\boldsymbol{D}$. I decided to try to extract eq. 4.27, which states:
$\...
- 259
2
votes
3
answers
106
views
When two perpendicular lines are rotated about each other (out of their common plane), what is the new angle between them?
When two perpendicular lines are rotated about each other (out of their common plane), what is the new angle between them?
More formally: Let $\ell, m$ be two perpendicular lines, and let their common ...
- 2,722
1
vote
1
answer
68
views
Why doesn't converting a difference quaternion to euler angles give angular velocity?
According to https://stackoverflow.com/questions/22157435/difference-between-the-two-quaternions
I can get the difference between two quaternions as
...
- 185
0
votes
0
answers
27
views
Matrix Representaions of Binary Icosahedral Group's Element
I am wondering how to write rotations of binary icosahedral group as matrices.
For example, the identity element of this group corresponds:
$R_1 = 1 + 0i + 0j + 0k$
since this group corresponds to ...
- 207
0
votes
0
answers
32
views
Knowing the equation of the tangent to a rotated curve, how to find the point of tangency.?
Desmos construction : https://www.desmos.com/calculator/uul5rbl6gd
Note : in the case of non rotated curves, the coordinates of the point of tangency can be " read off" on the equation of ...
- 1,597
1
vote
3
answers
111
views
Matrix of rotation in $\mathbb R^{3}$
I'm attempting to teach myself basic multivariable calculus and so far finding some of the 3d visualizations a bit difficult. Here's an example:
Let $f: \mathbb R^{3} \longrightarrow \mathbb R^{3}$ be ...
- 420
2
votes
1
answer
44
views
Generate a randomly-oriented vector $\vec{w}$ at a specific angle $\theta$ from another vector $\vec{v}$ in $\mathbb{R}^n$
I'm currently facing a problem where I need to perturb a provided vector $\vec{v}$ to produce a new vector $\vec{w}$, where both are located in $\mathbb{R}^n$, by perturbing both the magnitude and ...
0
votes
0
answers
15
views
"Canonically" rotate a path/trajectory/sequence of points
I have a sequence of $n$ points in $\mathbb{R}^3$: $$P_0, P_1, P_2, \ldots, P_n$$ where $ P_i = (x_i, y_i, z_i).$ We can assume, that they are "centered", i.e. the mass center (average) is ...
- 421
0
votes
2
answers
54
views
Prove that holomorphic functions preserve their norms under rotations.
I want to prove (or disprove) that any non-constant holomorphic function preserves its absolute value under rotations of its domain.
The precise formulation of the problem is as follows:
Let $f:\...
0
votes
0
answers
52
views
Algebraically showing 2 quaternions compose a new rotation
I am trying to algebraically prove that multiplying 2 quaternion rotations produces a new quaterion rotation. This is what I have so far:
After lots of simplifying, I get here:
I see the first term ...
- 101