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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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Prove the transfer between roll. pitch, yaw to tilt, azimuth, swing

I have a paper that explain the transfer between yaw, pitch, roll to the form of tilt swing, azimuth. The paper I need to prove that the following equations (from the paper) are true: Pitch first, ...
Idan Aviv's user avatar
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How to define $\mathbb{H}^2$-rotation about any point?

In the book Geometry of Surfaces by Stillwell, he defines the $\mathbb{H}^2$-rotation about $i$ in view of the conformal disc $\mathbb{D}^2$. Specifically, let $$ J(z):=\frac{iz+1}{z+i}$$ be the ...
Zoudelong's user avatar
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3 votes
2 answers
60 views

How to interpret the condition of a circumference rolling without slipping on another circumference

Suppose a circumference of radius r and center $\Omega$ rotates with constant angular velocity $\omega_D=\dot\phi e_3$ (D stands for disk) around an axis parallel to $e_3$ through $\Omega$. Let $\...
Davide Masi's user avatar
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0 answers
21 views

Spherical Polar Rotations | Is $(\theta + \pi, \phi + \pi) $ the same as $ (\theta + \pi, 0)$?

Question: I'd like some clarity on whether or not I can describe a $180^\circ$ rotation in spherical polar coordinates as $(\theta + \pi, \phi + \pi)$ affecting only the angles $\theta$ from the ...
Nathan Ngqwebo's user avatar
-2 votes
0 answers
30 views

Rodrigues (Gibbs) vector and the Cayley vector difference [closed]

Both vectors are axis of rotation with magnitude tangent half angle. gv = cv = axis * tan(angle/2) So why, in literature this entities are split? Both usually ...
minorlogic's user avatar
0 votes
1 answer
25 views

Why doesn't the rotation of a inertia tensor by a rotation matrix cancel itself out?

I'm currently reading about rotating an inertia tensor by a rotation matrix. I've found this formula for the rotation: $I'=RIR^T$ Where R is the rotation matrix and I is the inertia tensor. I can't ...
Laxen5's user avatar
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1 vote
0 answers
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Rotation angle between two parallel transport (Bishop) frames

I am conducting academic research on steerable needles with multiple sections and I have run into a roadblock. Each needle is composed of two or more sections connected in series and each section can ...
user1346036's user avatar
1 vote
1 answer
37 views

Rotation along Cartesian coordinates in terms of $\theta$ and $\phi$ of spherical coordinates

Suppose we have the following cartesian and spherical coordinates. And the rotation matrices $(R_x,R_y,R_z)$ along three cartesian coordinates follows the right hand rotation rule as following: Now ...
Luqman Saleem's user avatar
1 vote
0 answers
103 views

Linear approximation of the magnetic dipole field

Summary: using 3 angles to represent a magnetic dipole's orientation is redundant because the rotation around the $z$-axis of the dipole does not change the magnetic field, there are only 2 DOFs for ...
William Lin's user avatar
0 votes
1 answer
34 views

Generalized Rotational Matrix for n-dimensional Euclidean Vector Spaces [duplicate]

$R_{ij}(\theta) := \begin{bmatrix}I_{i-1} & 0 & ... & ... & 0 \\ 0 & \cos(\theta) & 0 & -\sin(\theta) & 0 \\ 0 & 0 & I_{j-i-1} & 0 & 0 \\ 0 & \sin(\...
nameless___'s user avatar
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1 answer
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How to calculate the clockwise rotation (bearing) from 3 known coordinates and independent of the cartesian XY axes

Greetings Maths experts, I come to you once more looking for mathematical assistance to help me solve another challenge in my CAD software. Background Info: I'm trying to write a VBA macro which will ...
SmartSolid's user avatar
3 votes
6 answers
230 views

Rotating and scaling an arbitrary triangle such that the new triangle has its vertices on the sides of the original one

Given $\triangle ABC$, and a scale factor $r \lt 1 $, I want to find the necessary rotation (center and angle) such that the rotated/scaled version of the triangle has its vertices lying on the sides ...
Quadrics's user avatar
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2 votes
3 answers
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How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?

$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$ $D$ is a ball of radius $R$ $\mathbf{r}$ is the position vector of the point where we want to ...
giannisl9's user avatar
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2 answers
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Why do we need quaternions when we already have continuous mappings $\mathbb{R}^3\to SO(3)$?

Forgive me for asking such a basic question. What are the advantages of using quaternions over the following continuous mapping $\mathbb{R}^3\to SO(3)$? The mapping $f:\mathbb{R}^3\to SO(3):v\mapsto ...
Pineapple Fish's user avatar
3 votes
2 answers
172 views

Rotating a given plane into another given plane

Suppose you're given two planes $ n_1 \cdot r = d_1 $ $ n_2 \cdot r = d_2 $ where $n_1$ and $n_2$ are unit vectors (known). $d_1$ and $d_2$ are known scalars. I want to rotate the first plane into ...
Quadrics's user avatar
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2 answers
110 views

Can any line be rotated such that it touches two arbitrary points?

Using one translation and a rotation this can be done. From the answers there, I assume that one needs the additional translation. But I can't think of an example where this is obvious. Edit: To ...
M0M0's user avatar
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1 answer
33 views

How can I adjust the x, y, and z coordinates of inscribed circles on a spinning sphere for different polar angles in an animation?

I am making an animation of a spinning sphere with circles inscribed on it. I have been successful in rotating it azimuthally. For polar angles, while I can rotate the inscribed circles appropriately, ...
Jasper's user avatar
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0 answers
19 views

uniqueness of axis-angle decomposition

I know that it takes three real numbers to specify an element of ${\rm SO}(3)$, which can be thought of as two numbers to specify an axis and a single number specifying the angle of rotation. However ...
proteus7's user avatar
1 vote
3 answers
132 views

How to find the equation of an ellipse using three points?

I came across this interesting problem yesterday and I am not quite able to find the equation of the ellipse after it has performed that roll. The original problem shows the ellipse to rotate till it ...
whatamidoing's user avatar
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0 answers
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How to convert to the Cartesian coordinates from new spherical coordinates after the rotation of spherical coordinate axes?

In a spherical coordinate system, an arbitrary point on the unit sphere is represented by the coordinates $(\theta, \phi)$. Although the $\theta$ and $\phi$ values are defined with respect to the $x$-,...
kachigusa's user avatar
  • 101
-1 votes
1 answer
71 views

How to calculate the difference between quaternions [closed]

I have written some code in python. The orientation of different objects in the simulation are stored using quaternions. At one point I have some orientation q and another orientation q'. I need to ...
rotatedBananna's user avatar
1 vote
1 answer
39 views

Question about Euler angles and rotation about relative and fixed frames?

I'm studying linear algebra, and one of the topics is rotation through euler angles. Depending on the sequence, we obviously get different results. One thing that I'm confused about however, is that ...
JerSci's user avatar
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1 answer
40 views

Matrices invariant under rotations are always proportional to the identity?

Is this proof true? Suppose we have a $3\times 3$ matrix $M^{ab}$ satisfying $$M^{ab}=R^a\,_cR^b\,_dM^{cd},$$ i.e. $$M=RMR^T,$$ for all rotations $R\in \mathrm{O}(3)$. Now, if denote representations ...
Ivan Burbano's user avatar
  • 1,258
6 votes
4 answers
340 views

Rolling an elliptical disc on the $x$ axis

You're given the elliptical disc bounded by $ \dfrac{x^2}{a^2} + \dfrac{(y - b)^2}{b^2} = 1 $ where $a = 5, b = 2 $. You roll this ellipse to the right along the positive $x$ axis, such that it is ...
Quadrics's user avatar
  • 24.3k
1 vote
1 answer
37 views

Find appropriate rotation matrix with non-square matrices

In linear algebra, consider:: $$\pmb{G}: L \times K$$ matrix $$\pmb{F}: T \times K$$ matrix $$\pmb{H}: T \times L$$ matrix. The apex $\intercal$ denotes the transpose. It holds that $T > L > K$ ...
user9875321__'s user avatar
0 votes
1 answer
26 views

Why does $R_{a,\theta}S_L(a)=R_\theta S_L(a)+(1-R_\theta)a$?

I'm working on a problem, Show that $R_{a,\theta}S_L(x)=T_c$, where $c=(1-R_\theta)(a-S_L(a))$. Here $R_{a,\theta}$ is rotation by angle $\theta$ about point $a$; $S_L$ is reflection in line $L$; $...
mjc's user avatar
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1 vote
1 answer
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Determining the Center of Rotation in a Video: A Mathematical Approach

I have a video where the camera rotates, causing the images to rotate around a specific point. I need to determine the coordinates of this rotation center. Here's my plan: I use a function to measure ...
Peter Jackson's user avatar
1 vote
2 answers
73 views

How do I get this $Q(x,y)$ into a sum of squares without matrices

The bivariate quadratic polynomial $Q(x,y)$ is: $$Q(x,y)=x^2+y^2+xy-a(2x+y)$$ to get it into a sum of squares, is there a method without any rotation of matrices involved? I can kind of can get it to ...
Ivy's user avatar
  • 87
2 votes
2 answers
64 views

How can I decompose a 3D rotation into one rotation about Z and one about an axis in XY?

How can I decompose a 3D rotation $T$ into one rotation about Z and one about an axis in XY, the latter with minimal angle? Note: This is similar but not identical to How can I break down a rotation ...
Jann Poppinga's user avatar
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0 answers
24 views

What is the derivative of unit quaternion time derivative w.r.t. to unit quaternion and angular velocity?

I am trying to get the Jacobian matrix of continuous-time rigid body dynamics using unit quaternions. The state vector is $x=\left[p, q, v, \omega\right]$. $p, v, \omega\in\mathbb{R}^3$ are position, ...
Atom's user avatar
  • 1
2 votes
1 answer
64 views

Proof rotation matrix is symmetric when Trace is -1

For a rotation matrix on SO(3), IE 3 dimensional, if the trace is -1 how do you prove it is symmetric? Intuitively it makes sense as this is 180 degree rotation but I don't see an obvious proof.
maxical's user avatar
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1 answer
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Question about Cartan's Theory of Spinors, Section 53 a spinor is a Euclidean tensor

Context I'm studying spinors in detail as part of research project. I'm working through Cartan's Theory of Spinors [1]. In section 53, A spinor is a Euclidean tensor, Cartan asks us to, "Consider ...
Michael Levy's user avatar
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0 votes
1 answer
50 views

Calculating the corners of a rotated outer rectangle that encapsulates minimally an inner rectangle.

I have two rectangles that start as the same size. When I rotate one of these rectangles I want it to encapsulate the other rectangle taking up the minimum possible area. The coordinates of the ...
Jfloaty's user avatar
  • 13
1 vote
1 answer
57 views

Deriving the Unit Quaternion to Tait-Bryan Angles conversion.

Let me start by saying I have a working solution. But I just don't understand how to get there. I've followed the well-written paper Technical Concepts Orientation, Rotation, Velocity and Acceleration,...
Michael Marcin's user avatar
1 vote
1 answer
43 views

Deriving the Finite Rotation Formula (Rodrigues's Rotation Formula)

I am working on Derivation 12 of Chapter 4 on p. 181 of the $3^{\mathrm{rd}}$ edition of Goldstein's Classical Mechanics. The question, paraphrased for readability, asks: In a set of axes where the $z$...
Georgy Zhukov's user avatar
-1 votes
1 answer
27 views

Combining rotations [closed]

Is there a way to get this rotation in 3D space with math (I assume with matrices)? Gif of said rotation
tvtomas's user avatar
-1 votes
1 answer
58 views

Determine the rotation necessary to bring an ellipse in contact (tangent) with another ellipse

Question: Given a fixed ellipse $ (r - C_1)^T Q_1 (r - C_1) = 1$ where $r=[x,y]^T $ , $C_1$ is the center, and $Q_1$ is a symmetric positive definite $2 \times 2 $ matrix. And a second ellipse $ (r - ...
Quadrics's user avatar
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0 votes
1 answer
51 views

Shell and disc method applied to a specific integral [closed]

After watching Khan Academy, Org Chem tutor and a few others, my understanding is this: whether you solve in terms of $x$ and $y$ entirely depends on which axis you work with. For shell - if you are ...
Anish Shah's user avatar
1 vote
1 answer
54 views

Find the angle of rotation to minimize the difference between the rotated vector and a given vector

Question: Given vectors $E$ and $F$ both of unit length, I want to rotate vector $E$ about a fixed known axis $a$ such that the rotated vector $E'$ has the minimum possible difference with vector $F$. ...
Quadrics's user avatar
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0 votes
0 answers
38 views

How to get the correct 3D rotation matrix given two vectors?

Context: Imagine there is a coordinate axes and a normalized vector, v1 from its origin. Now, suppose the coordinate axes rotates only by yawing a certain angle, $\phi$. After this rotation, the ...
Hritik RC's user avatar
4 votes
3 answers
113 views

Find the rotation necessary to bring a circle in contact with an ellipse

Question: Given an ellipse $ (r - C)^T Q (r - C) = 1$ where $r = [x,y]^T $ , $C$ is the center, and $Q$ is a $2 \times 2$ symmetric positive definite matrix. And a circle initially given by $ (r - r_0)...
Quadrics's user avatar
  • 24.3k
0 votes
2 answers
48 views

Determine the rotation necessary to make a line tangent to an ellipse

Question: Given the line $ \ell(t) = r_0 + t \ u $ I want to rotate it about point $P_0$, such that it becomes tangent to the ellipse $ (r - C)^T Q (r - C) = 1$ where $ r = [x,y]^T$, $C$ is the center ...
Quadrics's user avatar
  • 24.3k
1 vote
0 answers
30 views

I have two formulations of quadrotor dynamics, one in euler angle velocities, and one in body frame angular velocities. I am unable to see equivalence

In the following MIT lecture notes Equation (6.10) is on the following form \begin{equation}\begin{bmatrix} m \dot{v}^w \\ J \dot{\omega}^B \end{bmatrix} = \begin{bmatrix} -mge_3 \\ -\omega^B \times J ...
joh's user avatar
  • 11
0 votes
0 answers
47 views

Optimal rotation matrix

Context: I am trying to adapt the Rigid point set registration algorithm from Point Set Registration: Coherent Point Drift to include rotation information. My problem can be stated as follows: $\max_{...
MDescamps's user avatar
  • 161
1 vote
0 answers
44 views

Find appropriate rotation matrix

Consider the following elements in linear algebra: $\pmb{G}: L \times K$ matrix $\pmb{F}: T \times K$ matrix $\pmb{H} = \pmb{G} \pmb{F}^{\intercal}: T \times L$ matrix. The apex $\intercal$ denotes ...
user9875321__'s user avatar
1 vote
1 answer
20 views

Rotation by exchange of components

Consider the rotation matrices around e.g. the $z$-axis and the $y$-axis $$ R_z(\phi) = \left[ \begin{matrix} \cos \phi & - \sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 &...
Octavius's user avatar
  • 700
0 votes
1 answer
71 views

Describing the Eigenspace of a Linear Transformation on $\Bbb R^3$ that Rotates Points About a Line Through the Origin

Let A be the matrix of the linear transformation $T$. Without writing A, find an eigenvalue of A and describe the eigenspace in the following situations. a. $T$ is the transformation on $\Bbb R^2$ ...
Grey's user avatar
  • 723
2 votes
0 answers
93 views

Can we sequentially rotate a die on 3 axes from a given starting position so that the result is uniform?

Edit It seems that my initial assumption that the rotations in question occur simultaneously was wrong. They are calculated sequentially but animated simultaneously. Therefore I have updated the ...
pawello2222's user avatar
1 vote
0 answers
24 views

Effect of quaternion normalization when differentiating

I'm interested in optimizing across rotations which are represented as quaternions. You can either differentiate and then normalize after the update or you can include the normalizing term in the ...
maxical's user avatar
  • 603
4 votes
4 answers
713 views

The rotation symmetry group and the reflection group: Is there a name for what they have in common?

When I turn on my monitor, the brand name fills the screen. But since I mounted my monitor upside down so I can watch it in bed looking up, the power-on screen is upside down. But I noticed that it ...
Miss Understands's user avatar

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