# Questions tagged [rotations]

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

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### 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 ...
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### 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 ...
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### 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
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### 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
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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 ...
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### 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: $q(\theta, {\bf n}) = \left( \cos\frac{\theta}{2}, {\bf n} \sin \frac{\theta}{2} \right)$ ...
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### 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
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### 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
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### 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 $z$...
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### Combining rotations [closed]

Is there a way to get this rotation in 3D space with math (I assume with matrices)? Gif of said rotation
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### 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 ...
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### 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
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### 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{bmatrix} m \dot{v}^w \\ J \dot{\omega}^B \end{bmatrix} = \begin{bmatrix} -mge_3 \\ -\omega^B \times J ...
1 vote
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### 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 ...
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 ...
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### 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 ...