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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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0 votes
2 answers
30 views

How to calculate the difference between quaternions

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 ...
0 votes
0 answers
10 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 ...
0 votes
1 answer
25 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 ...
1 vote
1 answer
32 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 $T > L > K$ ...
1 vote
0 answers
36 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 ...
0 votes
2 answers
958 views

Converting local rotation to global rotation.

I have a rotation matrix in a local coordinate system and a 4x4 homogeneous matrix. I'm trying to convert the local rotation matrix to the global rotation matrix. I tried to find the dot product ...
0 votes
1 answer
22 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 $L$; $...
1 vote
1 answer
69 views

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 ...
2 votes
2 answers
59 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 ...
0 votes
0 answers
29 views

How can we apply the Lorentz transformation to perform consecutive boosts in different directions for multiple frames of reference? [closed]

thank you for coming to help, my problem is I need to do a Lorentz transformation for multiple frames as I suggest in the topic. What I have already done is find the Lorentz factor for each boosting ...
1 vote
2 answers
72 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 ...
-2 votes
0 answers
34 views

How to derive 3 dimensional rotation matrix [closed]

I was wondering how the rotation matrices for 3 dimensional matrix rotation was derived.
0 votes
0 answers
19 views

Unique affine transformation, sending $P_1$ to $P'_1$, $P_2$ to $P'_2$, $P_3$ to $P'_3$ where $\| P_i P_j \| = \| P'_i P'_j \| $

Given three points in $3d$ space $\{ P_i, i = 1, 2, 3 \} $, and corresponding three points $\{ P'_i, i = 1, 2, 3 \} $ that are their images, respectively, such that $\| P_i P_j \| = \| P'_i P'_j \| , ...
1 vote
1 answer
2k views

Find local coordinate system from rotation matrix (or quaternion) and a direction vector

In order to rotate body $B_2$ properly, I need to determine the local coordinate system (vectors of $x$, $y$ and $z$-axis) based on body $\frac{B_1}{B_2}$ and align the $z$-axis of this coordinate ...
94 votes
3 answers
5k views

Modelling the "Moving Sofa"

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
-1 votes
1 answer
55 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 - ...
1 vote
2 answers
106 views

I need an alternative 3d rotation/translation matrix for a pygame visualization.

Rotation around arbitrary axis ...
0 votes
0 answers
20 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, ...
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 ...
1 vote
1 answer
48 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 $F$. ...
1 vote
1 answer
184 views

Equation of hyperboloid of one sheet resulting from rotating a (skew) line about an axis

Suppose I have a line in $3D$ given in parametric form as $ L(t) = P_0 + t V $, with $t$ being the parameter, and I rotate it about an axis passing through the origin whose direction is specified by ...
2 votes
1 answer
58 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.
1 vote
1 answer
2k views

How to move and/or rotate a semi circle or like any equation. Or just add a perimeter cut off on one of the sides.

I'm just some random 6th grader having fun in graphing trying to get some practice in before high-school but my teacher says she doesn't know how to rotate any equation or how to rotate this semi ...
3 votes
1 answer
157 views

What $3 \times 3$ matrix gives the determinant $a^{2}+b^{2}+c^{2}$?

The $2D$ rotation matrix is given by a $2\times{2}$ matrix: $$\begin{bmatrix} a & -b\\ b & a \end{bmatrix}$$ The determinant of this matrix is $a^{2}+b^{2}$. Can one construct a $3\times{3}$ ...
2 votes
1 answer
1k views

Rotation of a circle over another circle

Let the circle $d$ rotates over the circle $c$. If the angle rotation to be of 0 to 360, then how many times does the circle $d$ rotate during that process? Since the total external angle is 360, I ...
0 votes
1 answer
39 views

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 ...
0 votes
1 answer
47 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 ...
1 vote
1 answer
52 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,...
0 votes
2 answers
47 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 ...
5 votes
2 answers
1k views

Fixed and current axes of rotation

I was astonished by ingenuity of many users who demonstrated reasons for why rotational matrices are not commutative. However in 3d rotations I'm more puzzled by some other theorem ... How ...
4 votes
3 answers
107 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)...
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: $q(\theta, {\bf n}) = \left( \cos\frac{\theta}{2}, {\bf n} \sin \frac{\theta}{2} \right)$ ...
0 votes
1 answer
49 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 ...
1 vote
2 answers
3k views

Orientation of a 3D plane using three points

I have a 3D plane and three points on this plane with known coordinates of these three points. How can I find the orientation of this plane i.e. angles of this plane with X, y and Z-axis.
1 vote
1 answer
34 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 $z$...
-1 votes
1 answer
25 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
2 votes
2 answers
1k views

Find bounding box dimensions around rotated object

Consider the following rectangle with dimensions 320 by 130. After rotating the rectangle 10 degrees clockwise from the center (x: 160, y: 65), it looks like this. My question is: How do I ...
0 votes
0 answers
37 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 ...
1 vote
0 answers
29 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 ...
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 ...
2 votes
1 answer
71 views

Why are rotation numbers not homomorphic?

If $f,g$ are degree-1 monotone maps of the circle, why do we generally have $\rho(f\circ g)\neq\rho(f)+\rho(g)$? I mean, you might say that we have no right to expect an equality. After all, it's not ...
0 votes
0 answers
46 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_{...
2 votes
1 answer
298 views

The parametric equation of the shortest curve along a sphere between any 2 points on sphere

So, I am trying to find the parametric equation of the shortest curve on a sphere between two arbitrary given points on a sphere. Starting from cartesian coordinate system, My plan so far is, find ...
1 vote
1 answer
18 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 &...
0 votes
1 answer
55 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 $\Bbb R^2$ ...
2 votes
1 answer
1k views

How do I rotate vector? [closed]

It's given vector $\vec{e_{1}}=0.31\vec{e_{x}}+0.95\vec{e_{y}}$. How do I rotate that for 30 degrees counterclokwise? What I have done, I have used $x’=x\cos\theta-y\sin\theta$, $y’=x\sin\theta+y\cos\...
2 votes
0 answers
90 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 ...
2 votes
1 answer
1k views

Velocity vector from inertial to body frame

here's my question: I have position and velocity vectors of a body in the inertial frame. Now i need to switch the reference system to body frame. So i have $\bar{x}_b = \hat{R}\bar{x}_I$ where $\hat{...
1 vote
1 answer
92 views

Is it possible to extract the translation of an affine transformation matrix independent of rotation center and angle?

I have $2$ images rotated by $60^\circ$ to each other with different center of rotation. The here presented matrices are affine transformation matrices derived from OpenCV: https://docs.opencv.org/4.x/...
0 votes
0 answers
14 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 ...

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