# 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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### 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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