Questions tagged [rotations]

This tag is for questions about *rotations*: a type of rigid motion in a space.

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Transforming a permutation into a rotation

Let $n\geq 3$ and $G=C_{n}=\{1,r,...,r^{n-1}\}$ be the cyclic group of $n$ elements where $r$ is the rotation of $360/n$ degrees. Here, let us consider a vector $x\in\mathbb{R}^{n}$ as consisting of ...
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Calculate rotation of phone for steering based on pitch and roll

I am creating a mobile game that uses the orientation of the phone to steer. I have access to the pitch and roll of the device relative to the Earth. I currently use the roll of the device to change ...
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1answer
67 views

SVD - Finding the angle of rotation from U and V

Given a 2×3 matrix, the Singular Value Decomposition would give the matrix U which would be a 2x2 matrix and VT (transpose of V), a 3x3 matrix. From what I ...
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2answers
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n Dimensional Rotation Matrix

So the rotation matrix for 2D is: $\begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix}$ and one of three BASIC rotation matrices for 3D is: $\begin{...
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1answer
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Geometric rotations of a group of objects

Referring to the first diagrams I am trying to copy the three objects, looking at them from an arbitrary angle(A1). The distance between where I am and the first object does not matter just the ...
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1answer
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Volume by Rotation Using Integration

How to find the volume if the shown area is rotated around the $y$-axis? The area will be bounded by $𝑦=𝑥^2+1$, $y=2x$ and $x=0$.
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1answer
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Entire function that is a bijection on the unit disk is a rotation

I'm working on this problem "Let $f$ be an entire function. Suppose $f$ restricted to the unit disk is a bijection. Prove that $f$ is a rotation." My attempt: It is tempting to use Schwarz lemma. ...
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1answer
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Any reference(a book) that defines the $n$-dimensional rotation matrix?

I want to refer to a mathematics book that explains the n-dimensional rotation matrix or rotation transformation. Wikipedia concentrates most on 2D or 3D. There are things that one can say definition ...
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Optimization solver, Rotation Matrix

As a part of my work I want to find a transformation matrix from Frame A to Frame B. Let us define Frame A axis ($X_a,Y_a,Z_a$ - all are orthogonal to each other) Frame B axis ($X_b,Y_b,Z_b$ - all ...
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Determining attitude from a single set of vectors.

I'm given a large list of absolute points in 3D space, as well as vectors to a subset of these points from a single unknown location, collected from something like a star tracker. I'm tasked with ...
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Varimax-Rotation in Factor Analysis

So I know that the orhogonal Factor Model is constructed as such: $X-\mu=LF+\epsilon$, where $\text{Cov(X)=}\Sigma=\text{LL'}+\Psi$. Then the loadings $L$ can be constructed from the square root of ...
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1answer
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Isometries: Find the vectors and matrices

We consider the isometries of $\Bbb R^2$. Let $\varphi$ be the rotation of $90^\circ$ (counterclockwise) around the point $\begin{pmatrix}3 \\ 5\end{pmatrix}$ and let $\psi$ be the reflection about ...
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Rotating the Sides of a Quadrilateral Rotates its Diagonals

Rotating the 4 sides: Given ABCD, rotate AB by $\theta$ to get A'B', and rotate CD by $\theta$ to get C'D'. (When we speak of rotating a segment by $\theta$, we mean rotating it by an angle of $\...
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Symmetries of Rotation matrices

Consider the discrete symmetry $C_3$ (rotations by $120^\circ$ leave system invariant). I believe then the matrix describing this action is simply the rotation matrix (in 3d) (clockwise rotation): $$...
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1answer
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Change in a quantity described by a dot product

In the Rotations section of Arfken's chapter on vector analysis in the book "Mathematical Methods for Physicists", there is a statement which says the following - The dot product is the projection ...
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Find rigid body transform $T_b$, given transforms $T_a, T_c$, where $T_c = T_b^{-1}T_aT_b$ for $T_b$. i.e. find change of basis.

Let $\mathbf{T}_a, \mathbf{T}_b, \mathbf{T}_c \in SE(3)$ be rigid body transforms, and; $$ \begin{equation} \mathbf{T}_c = \mathbf{T}_b^{-1}\mathbf{T}_a\mathbf{T}_b. \label{eq_basis} \end{equation} $...
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Keep relationship between trajectories each with their own coordinate system

I am currently estimating a robot's pose (3d position and rotation matrix) with an IMU and I want to reset the localizer in the middle of the run while keeping the relationship to the pose it had when ...
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1answer
28 views

Rigid body moving orientation relative to helical trajectory

A rigid sphere moves on a trajectory, made up of points that come out of a mechanical simulation. To a visual inspection, the trajectory is helical. I calculate the axis $\boldsymbol{N}$ by using ...
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1answer
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For rotation matrices in SO(3), solutions of AB = BA given A (or B)

Suppose we have $\mathbf{AB} = \mathbf{BA}$, where $\mathbf{A},\mathbf{B} \in SO(3)$. What facts does this imply about $\mathbf{A}$ and $\mathbf{B}$? Clearly $\mathbf{A} = \mathbf{I}_3$ and $\mathbf{...
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1answer
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How to calculate the inertia tensor for a cuboid with non-uniform mass distribution?

I've posted a similar question here yesterday that I though would solve my problem but I don't think it fully encompasses the problem I'm having so I'm posting a new question (I do think it's a ...
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1answer
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Rotation matrix and unitary similarity

Let $$ \text{U}(\theta;i,j)= \begin{pmatrix} 1 \\ & \ddots \\ && 1 \\ &&&\cos\theta&&&&-\sin\theta \\ &&&&1 \\ &&&&&\ddots \\...
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Reducible representation of $SO(2)$ on $\mathbb R^3$

A representation of $SO(2)$ on $\mathbb R^3$ is the the map $$\begin{bmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{bmatrix} \rightarrow\begin{bmatrix}\cos\theta & \sin\...
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Local and global rotation inside hierarchy using quaternions

When I want to rotate a Model using Quaternions I have to either left or right multiply the current orientation with the rotation quaternion: ...
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Rotational invariance of functions

What is the difference between a) $\mathbf{f} \rightarrow \mathbf{f}(\mathbf{R}^T\cdot\mathbf{x})=\mathbf{f}(\mathbf{x}^\prime)$, (Wikipedia, the article about rotational invariance) and b) $\...
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1answer
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rotational symmetries of the 120-cell

I want to find the number of rotational symmetries of the 120-cell but I am not very familiar with polytopes nor counting symmetries. So, I don't know if someone can give me an idea or an example with ...
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Different descriptions for $R^3$ and its rotations from the Lie Algebra of $SO(3)$

I'm a student of physics who just started studying group theory. I've tried to grasp the geometrical meaning of some things i've studied and I came up with an intuition. Can you tell me if it's ...
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1answer
32 views

Looking for a simple explanation of Singular Value Decomposition in practice

tl/dr: I'm trying to find the best rotation between two 3d point clouds, and all the answers say "use SVD", but I don't have the math background. However, once I get the concept, hopefully I can use ...
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$SO(3)$ irreducible representation and rotations

There are many irreducible representations of $SO(3)$ on $\mathbb R^3$. Consider the irreducible representation consisting of the map between the elements of $SO(3)$ to the linear transformations ...
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1answer
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Is there a rotation matrix that can invert any 3D vector? [closed]

For example, could a single rotation matrix convert the following vectors: vec{1, 1, 1} to vec{-1, -1, -1} ...
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Questions About Complex Rotation

This animation shows that the infinite sum of vector tips representing the Maclaurin Series of $e^{i\theta}$ lies on the unit circle in the complex plane for any value of $\theta$. The first vector ...
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Derive a rotation matrix to partial derivative

I have the following rotation matrix: [u = [cos θ -sin θ| [x v] = |sin θ cos θ] y] the rotation matrix is 2x2 (I apologize for my bad html skills) And I need to show that the ...
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Removing the rotational component in optical flow

I am working on a basic self driving algorithm using a monocular image sequence. For this, the optical flow between every two frames based on tracked keypoints is calculated (which is a vector for ...
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1answer
30 views

Which is the rotated frame in which the given matrix transforms into this block matrix? [UPDATED]

Given a matrix $A=\begin{pmatrix} a & b & c \\ b & a & -c \\ c & -c & d \end{pmatrix}$ with positive $a,b,c,d \in R$ , which are the angle $\theta$ and rotation axis that ...
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2answers
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which of the following are the rotation matrices? [closed]

which of the following are the rotation matrices? I know that if det|matrix|=1 or M.Mt=I=Mt.M is the property that makes some matrix a rotation matrix. but in this case options (1,3,4) seems to be ...
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A curious interrelationship between distinct embeddings of $SO(M+1)$ into $SO(2M+1)$

The following seems to be a property of $SO(2M+1)$ for an arbitrary integer $M$, although I have not yet been able to prove it. (I can prove it for, e.g., $M=1$, and have numerically checked it for ...
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0answers
28 views

$SU(2)$ representation that conserves the Lie algebra

I've seen that there are many representations of $SU(2)$ on $\mathbb R^3$, one of these representation is the map from $SU(2)$ to the set of matrices of $SO(3)$. This representation conserves the Lie ...
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0answers
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Let $G$ be the set of the 6 axial symmetries and the 6 axial rotations of an hexagon $E$ that transform $E$ into itself

Let $E$ be a hexagon whose side's measure is $1$ and let $T$ be the set of all the triangles whose vertices are three of the vertices of the hexagon. Then, let $G$ be the set of the six axial ...
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1answer
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Rotation of a Plane in 3d about a line [closed]

I have a set of points in the XY planes, I want to translate them all to the YZ plane by rotating about a line. Basically, Here, in this cube Imagine I have a list of points on 2376 plane, I want to ...
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Rotation of co-ordiante axes for a cubic crystal: Matrix rotation

When carrying out a rotation of the co-ordinate axes, x1 x2 x3 to say x1' x2' x3', When dealinf with the rotation of axes about a vector (using an orthogonal rotation matrix R), I was taught to use ...
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1answer
36 views

Constructing an orthogonal rotation transforming a basis vector into the equal sum

Could anyone please help me to construct an orthogonal matrix which would rotate the first basis vector $v_1$ from an orthonormal set $\{v_k\}_{k=1}^N$ into a vector having equal projections on all ...
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1answer
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Rotation matrix $R$ on orthonormalbasis of $\mathbb{R}^2$ [closed]

I'm working on this problem and have no clue how to solve it. Any idea, hint or solution would be very appreciated. Let $E$=$\mathbb{R}^2$, equipped with the standard scalar product. Let $R_\theta$ ...
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0answers
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Choosing the pivot of a rotation matrix in the Similarity Transformation

I have arrived at a equation in the Similarity Transformation - $M_r$ = $T. M_{r-1}. T^t$ ,where T is the rotation matrix and $M_r$ ,$M_{r-1}$ are similar matrices. My aim is to find the rotation ...
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1answer
24 views

Few forces applied to body with shifted center of mass. How to find resulting force and torque at central point?

I have body built from few 2d polygons each of them has mass. Forces can be applied to different points of body. How to calculate resulting force and torque at the central point of body (point (0;0) ...
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A slight modification on Banach-Tarski Paradox about $F_2<SO(3)$

It is known that proving $F_2<SO(3)$ is a key point on the way of demonstrating Banach-Tarski Paradox. So take two perpendicular rotations $$\left(\begin{matrix}\cos x&-\sin x&0\\\sin x&...
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How to rotate a Vec4 over Earth

I am working with WorldWind Java. I want to rotate a Vec4 on x,y and z-axis on Earth's Sphere. I found a function calculating the north pointing tangent. My question is how can I rotate this vector. ...
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How to find rotation matrix of a plane curve of varying curvature?

I'm able to find a rotation matrix with respect to a fixed basis for a plane curve of constant curvature (example - circle) or a straight line (zero curvature). But in the case of a sinusoid, ...
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1answer
27 views

How to translate, rotate and scale a meshgrid between two points in 3D space

I am trying to translate, rotate and scale a meshgrid between two points in 3D space, however, I cannot seem to get it right. I need my meshgrid to follow the points in order to make an animated 3D ...
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30 views

N-dimensional vector rotation

I am interested in rotating an N-dimensional vector v by a specified angle , i know about the Rodriguez method The thing is that my requirements do not allow for the presence of an orthogonal vector ...
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0answers
30 views

Finding Position, Rotation, & Speed. Moving Forward and Turning

I'm trying to figure out where my position and rotation would be while moving forward and turning. I'm hoping someone can point me in the right direction. Some of this i can figure out, but putting it ...
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error measure for unknown true rotation

I have some vectors $A$ and $B$ shape ($3 \times N$), and I approximate a rotation matrix $R$ so that $B\approx RA$ (in a least squares way). It's easy to compute the error of that approximation if I ...

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