# 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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### 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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### 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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### 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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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\... 0 votes 1 answer 17 views ### 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 |... 0 votes 1 answer 39 views ### 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 ... 0 votes 1 answer 28 views ### 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 ... 0 votes 0 answers 19 views ### 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 ... 17 votes 5 answers 2k views ### 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 ... 0 votes 1 answer 22 views ### 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 ... 1 vote 0 answers 16 views ### 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 ... 2 votes 2 answers 45 views ### 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}|)... 4 votes 1 answer 137 views ### 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 ... 1 vote 2 answers 34 views ### 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, ... 1 vote 1 answer 31 views ### 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 ... 1 vote 1 answer 64 views ### 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' $$for n = 2, one pair ... 0 votes 0 answers 5 views ### Deriving euler angles for the center of a rigid body based on euler angles from a point on that rigid body. I am in possession of a PCB that is capable of measuring it's orientation in terms of roll, pitch, and yaw, but unfortunately the person who made it neglected to make it so that these values are ... 1 vote 0 answers 25 views ### Understanding permutations of S_4 which correspond to symmetries of a regular tetrahedron I'm trying to understand values of the character of the representation of S_4 corresponding to the symmetries of a regular tetrahedron (whew, that's a mouthful!). One illustrative video is found ... 0 votes 1 answer 33 views ### Rotation identity proof In my script we took the following identity:$$\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}(...
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 ...
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\...