Questions tagged [rotations]

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

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ODEs to calculate the rate of quaternions knowing the angular velocities

A preamble: I know to convert quaternion in Eulerian Angles, I have to know the rotation orders adopted in that situation (i.e ZYX, or 321, YZX, or 231). Now, let's assume: ...
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I have 2 coordinate systems defined by 3 normal vectors each. How can i find the rotation matrix between the two, and euler angles?

I have a 2 coordinate systems defined using unit vectors. The first is the global cartesian csys1 = [[1,0,0],[0,1,0],[0,0,1]] The second is rotated by 90 degrees twice: csys2 = [[0,0,1],[1,0,0],[0,1,0]...
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Decompose a $2\times 2$ matrix to a combination of rotation matrices

The background I encounter this problem when I try to analyze the planar transformation of a 2D triangle. We ignore the translational shift in this problem. Consider a 2D triangle whose edge vectors ...
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Fractional Fourier Transform of $\sqrt{c} x( c(t - \tau))$

I am trying to figure out what the Fractional Fourier Transform of the signal $\sqrt{c} x(c(t-\tau))$ would be with respect to that of $x(t)$. According to the paper "The Fractional Fourier ...
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Generate a uniformly sampled orthonormal matrix that 'rotates' $k$ vectors $x_0 \in \mathcal{R}^{n \times k}$ into $y_0 \in \mathcal{R}^{n \times k}$

We know that orthonormal matrices $H \in \mathcal{R}^{n \times n} $ are rotation matrices. Is there a general method to uniformly generate rotation matrices that can rotate a given set of vector $x_0 ...
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Rotating about non-centered axis in 4-d

In 3 dimensions, there is a concept of rotating about an arbitrary line that is not centered at the origin. To pull this off, we first move the origin so that it is on this line. Anywhere on the line ...
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Counterpart of axis-angle rotation matrix in 4 dimensions?

In 3-dimensional space, we have an explicit formula for the rotation matrix which will rotate about a vector $\vec{a} = [a_x, a_y, a_z]$. This is given by: $$ \begin{bmatrix} \cos\theta+a_x^2(1-\cos\...
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General form of rank $2$ tensor invaiant under certain rotations

I have the problem of determining the most general form of a rank 2 tensor $t_{ij}$ (in 3 dimensions) satisfying: $t_{ij}$ is invariant under any rotation about the $z$-axis $t_{ij}$ is invariant ...
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Rotations in 3D space: order of concatenated rotations.

I am currently doing an online course in Udemy to clarify some things for a project I'm currently doing on solid rigid motion and simulation in University, although understanding some of the things ...
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Recovering matrix from rotated version

I'm dealing with matrices that came from a software, 3ds Max. It uses 4x3 matrices to represent transformations https://documentation.help/3DS-Max/idx_AT_matrix_representations_of_3d.htm $$\begin{...
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Is a map which sends a $3\times 3$ symmetric tensor to an element of $SO(3)$ which diagonalizes it necessarily discontinuous?

For a $3\times 3$ symmetric matrix $Q$, one can construct a map to $SO(3)$ which sends $Q$ to a matrix which diagonalizes it. If $Q$ has distinct eigenvalues, there are three choices for rotation ...
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how to offset a rotation contained in a unit quaternion rotation from the origin of a rigid object.

I'm using Unity3D for a project. The way it handles sorting transformations is with a 3vector-unit quaternion-3vector "sandwich" (the 1st vector for position, the quaternion for rotation, ...
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Why would there be any gimbal lock in 3D rotations?

When rotating EULER way, we rotate a matrix step by step right ? I tried in blender rotating Z 90° (could work with any axis) I can then rotate X&Y fine with no lock while a gimbal has its axis ...
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Quaternions for coordinate frame rotation

I am attempting to switch over from Euler matrices to quaternions. Normally I am presented with a desired final orientation vector in the form of a unit direction and the angles of rotation to create ...
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Map input from one vector space to another

I am working on a 3d animation project, where I get 3d coordinates of meshes of a face. Using these coordinates I want to move a different 3d face, which is different from the input face with respect ...
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How to represent the relative geometry of two ellipses with a common focus in GeoGebra?

I'm studying an astrodynamics problem and to help my study I'd like to represent the geometry I'm dealing with. I also obtained a figure in Matlab but I need to represent many angles and so I'd like ...
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To which angle do the wheels have to turn relative to their current position in order to turn correctly toward the object?

I am trying to write an algorithm to turn the EV3 robot (https://d2nmr6p48f8xwg.cloudfront.net/content_pictures/pictures/000/001/570/814c51fb41fab7a3e3039ec6a067accc510a9341Lego-Mindstorms-Ev3-Car-...
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Inverse of Euler parameter rotation matrix formulation [Barfoot 6.6.2

Reading through Barfoot's State Estimation for Robotics and I am curious regarding one of the problems: Show that $C^{-1}=C^{T}$ starting from $C = \cos\theta I > (1-\cos\theta)aa^{T} + \sin\...
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On the uniqueness of SVD in the $2$-dimensional case

On page 155 of Tristan Needham's Visual Differential Geometry and Forms, the singular value decomposition (SVD) is given by $$M = R_{\phi} \circ \Sigma \circ R_{-\theta}$$ with the associated picture: ...
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Property of hyperbolic rotation matrix with entry 1

I am considering the group of hyperbolic rotation matrices $G=\{A\in M_{3\times 3}(\mathbb{R}): A^TDA=D \}$, where $D=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&-1\\ \end{bmatrix}$. ...
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Rotations preserving a collection of linear subspaces

My question is the following: what is the subgroup of $O(d)$ that preserves a given collection of linear subspaces $V_1, \ldots, V_n\subset \mathbb{R}^d$? For a single subspace $V\subset \mathbb{R}^d$,...
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Making Sine Waves with Pointed Peaks

I am working on my own math project of movement around a square. (I am stuck) I have never been taught this stuff before(I'm only 15). So to start with I made a 4 unit by 4 unit square. I started by ...
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Calculating the rotation point between 2 sets of points

I've been trying to solve this for a while but I can't figure it out. I have 2 sets of points: p1, p2 and p1', p2'. p1(x1,y1), p2(x2,y2) and p1'(x1',y1'), p2'(x2',y2') all known values. Those 2 sets ...
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Which orthogonal rotation matrices for diagonalisation

We consider 2D orthogonal rotation matrices $R$. We consider a real matrix $$A = \begin{pmatrix} a & b\\ b & c \end{pmatrix}.$$ I write that $B = RAR^T$ for some diagonal matrix $B$. I would ...
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Trying to find a nice basis to realize the quaternion mapping as a rotation matrix

Trying to find a nice basis to realize the quaternion mapping as a rotation matrix. The quaternions are a 4 dimensional division algebra over $\mathbb{R}$ where we label the standard basis vectors as $...
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Create a circle perpendicular to a vector in a 3D coordinate system

I have been working on this problem for a few days already, but could not figure it out completely. Let's assume that I have two points, pointO (0,0,0) and pointA (1,1,1). Then we can calculate the ...
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Path of the sun across the sky in a 4D world

Someone asked a question on worldbuilding about navigating by the stars on a 4D planet. In thinking about it I came up with a question that seems appropriate to ask here, as it's purely a maths ...
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Angle between vectors after rotating one of them

I have a unit vector $\hat{a}$ and a unit vector $\hat{k}=\{\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\}$. Now I can rotate $\hat{k}$ such that it aligns with the z-axis, i.e. $\hat{k} \...
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Inverse of a roto-translation matrix in 3D space

I want to create two roto-translation matrices. The first transforms point $P$ into point $P'$ by performing a translation $T=(x_t, y_t, z_t)$ and two rotations (one around the $x$ axis of $\alpha$ ...
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Volume generated by rotating axis $X =2$ and the lines $x =0$ and $y = 1$

The question I wrote the shaded area equation and add 2 to it because it has shifted two units to the right, however the answer is wrong, where did I go wrong?
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Possible to find rotation around x, y, z axes by knowing the polar angles $\phi, \theta$?

I'm working with a 3D cartesian system $\vec{e'_x},\vec{e'_y},\vec{e'_z}$ that moves around in a global coordinate system. I know the origo position and $\vec{e'_z}$ in global coordinates. There is no ...
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1 vote
2 answers
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Find theta given point on circle

I am having trouble visualizing and understanding how you might obtain an angle given a point on a circle. I have a $(x, y)$ point where the values range between $0,1$ for both $x,y$. How would I ...
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How should I differentiate vector of $~\operatorname{rot}~$?

The following stuff handles a system where no electric charge exists(i.e. free space). $$\operatorname{rot}\boldsymbol{H}_{}=\sigma\boldsymbol{E}_{}+\epsilon{\partial\boldsymbol{E}_{}\over\partial\...
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Rotation Invariance of the Distribution of $\ell_2$-norm square of Gaussian Vector

In a paper I'm reading I saw the argument below: Let $G\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ and $A$ is a square matrix with SVD $A=U\Sigma V^T$ where $\Sigma=diag(\sigma_1,\dots,\sigma_n)$. Then $|...
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What is the rule for rotations about a point not the origin?

I know the rules for $90^\circ$(counterclockwise and clockwise) rotations, and $180^\circ$ rotations, but those are only for rotations about the origin. What is the rule for a rotation above that is ...
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A horizontal beam is attached to a wall by a cable. Find the force of the cable and the force exerted by the wall.

A 5.00m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of $53^o$ with the ...
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Rotating a vector to make it parallel to the z axis

I have to rotate the $P=(P_x,P_Y,P_z)$ point so that the $OP$ vector is parallel with respect to the $Z$ axis. To do this I perform 2 rotations, the first around the $Z$ axis and the second around the ...
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17 votes
5 answers
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Why can't rotations be represented by purely imaginary quaternions?

I imagine this question has a straightforward answer, but I haven't been able to think of it on my own. It's well-accepted that you can represent a rotation in three-space with a unit quaternion. In ...
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Rotation of a frame reference with axis z lying on normal n

I am creating a rotation matrix capable of converting the reference system $A=(x_a, y_a, z_a)$ into a second reference system $B=(x_b, y_b, z_b)$ where the $x_b$ and $y_b$ axes lie in a plane with ...
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Rotate a Translation and a Quaternion around the z axis of an arbitrary pose by an angle theta

I need to implement a rotation in a program but it's 15 years I haven't used rigid body motion maths. I use poses that are described by a translation T and quaternion Q. Everything is expressed in the ...
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Computing Euler angles between two 3D points from Cartesian coordinates

We are given the three-dimensional cartesian coordinates of a point $A$, a point $B$ and a point $C$. The distance from $A$ to $B$ is the same as the distance from $A$ to $C$ ($|\vec{AB}| = |\vec{AC}|)...
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How far does one have to zoom into an image that was rotated a certain amount of degrees in order to only see only the original pixels again?

This question was asked by a work colleague of mine, but my days as a mathematician are long gone unfortunately. It does sound like a pretty basic geometry problem to me, doesn't it? I'm not expecting ...
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What is the smallest 3D rotation to make the axes line up

A 3x3 rotation matrix is considered axis-aligned if it consists of only 1, -1, and 0. Given an arbitrary rotation matrix, what is the smallest rotation required to make it axis-aligned? For example, ...
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1 vote
1 answer
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How to decompose matrix to its composition of rotation and scaling?

Ans: for rotation and scaling factor are $\frac{\pi}{6}$ and 8 respectively. I found a related question, but it wasn't explaining it well. I also understand scale and rotation separately but cannot ...
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How many pairs of points at least on the unit sphere can be used to determine a rotation matrix in $\mathbb{R}^n$ uniquely?

A rotation matrix map the unit sphere $\mathbb{S}^n$ onto itself, we want to identify the rotation matrix $R$ by pairs of points $x,x'\in \mathbb{S}^n$, such that $$ Rx = x' $$ for $n = 2$, one pair ...
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Deriving euler angles for the center of a rigid body based on euler angles from a point on that rigid body.

I am in possession of a PCB that is capable of measuring it's orientation in terms of roll, pitch, and yaw, but unfortunately the person who made it neglected to make it so that these values are ...
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Understanding permutations of $S_4$ which correspond to symmetries of a regular tetrahedron

I'm trying to understand values of the character of the representation of $S_4$ corresponding to the symmetries of a regular tetrahedron (whew, that's a mouthful!). One illustrative video is found ...
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Rotation identity proof

In my script we took the following identity: $$\mathcal{R}_{\vec e_y}(-d\alpha')\mathcal{R}_{\vec e_x}(d\alpha)\mathcal{R}_{\vec e_y}(d\alpha')\mathcal{R}_{\vec e_x}(-d\alpha)= \mathcal{R}_{\vec e_z}(...
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2 votes
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Angular speed of an rotating ellipsoid

Consider an ellipsoid described the equation in cartesian coordinates $(x,y,z)$ $$ \frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{c^2} = 1, \quad \text{where} \quad a < c.$$ This ellipsoid is ...
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Infinitesimal rotation around an arbitrary axis S0(3)

In my script, I am reading about the case of a small infinitesimal rotation and it's approximation. If $R$ is the geometrical rotation, and we consider a vector $\vec{OM}$,an infinitesimal angle $d\...
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