# Questions tagged [rotations]

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

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### Prove the transfer between roll. pitch, yaw to tilt, azimuth, swing

I have a paper that explain the transfer between yaw, pitch, roll to the form of tilt swing, azimuth. The paper I need to prove that the following equations (from the paper) are true: Pitch first, ...
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### How to define $\mathbb{H}^2$-rotation about any point?

In the book Geometry of Surfaces by Stillwell, he defines the $\mathbb{H}^2$-rotation about $i$ in view of the conformal disc $\mathbb{D}^2$. Specifically, let $$J(z):=\frac{iz+1}{z+i}$$ be the ...
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### How to calculate the clockwise rotation (bearing) from 3 known coordinates and independent of the cartesian XY axes

Greetings Maths experts, I come to you once more looking for mathematical assistance to help me solve another challenge in my CAD software. Background Info: I'm trying to write a VBA macro which will ...
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### Rotating and scaling an arbitrary triangle such that the new triangle has its vertices on the sides of the original one

Given $\triangle ABC$, and a scale factor $r \lt 1$, I want to find the necessary rotation (center and angle) such that the rotated/scaled version of the triangle has its vertices lying on the sides ...
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### How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?

$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$ $D$ is a ball of radius $R$ $\mathbf{r}$ is the position vector of the point where we want to ...
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### Determining the Center of Rotation in a Video: A Mathematical Approach

I have a video where the camera rotates, causing the images to rotate around a specific point. I need to determine the coordinates of this rotation center. Here's my plan: I use a function to measure ...
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### How do I get this $Q(x,y)$ into a sum of squares without matrices

The bivariate quadratic polynomial $Q(x,y)$ is: $$Q(x,y)=x^2+y^2+xy-a(2x+y)$$ to get it into a sum of squares, is there a method without any rotation of matrices involved? I can kind of can get it to ...
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### How can I decompose a 3D rotation into one rotation about Z and one about an axis in XY?

How can I decompose a 3D rotation $T$ into one rotation about Z and one about an axis in XY, the latter with minimal angle? Note: This is similar but not identical to How can I break down a rotation ...
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### What is the derivative of unit quaternion time derivative w.r.t. to unit quaternion and angular velocity?

I am trying to get the Jacobian matrix of continuous-time rigid body dynamics using unit quaternions. The state vector is $x=\left[p, q, v, \omega\right]$. $p, v, \omega\in\mathbb{R}^3$ are position, ...
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### Proof rotation matrix is symmetric when Trace is -1

For a rotation matrix on SO(3), IE 3 dimensional, if the trace is -1 how do you prove it is symmetric? Intuitively it makes sense as this is 180 degree rotation but I don't see an obvious proof.
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### Question about Cartan's Theory of Spinors, Section 53 a spinor is a Euclidean tensor

Context I'm studying spinors in detail as part of research project. I'm working through Cartan's Theory of Spinors [1]. In section 53, A spinor is a Euclidean tensor, Cartan asks us to, "Consider ...
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### Calculating the corners of a rotated outer rectangle that encapsulates minimally an inner rectangle.

I have two rectangles that start as the same size. When I rotate one of these rectangles I want it to encapsulate the other rectangle taking up the minimum possible area. The coordinates of the ...
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### Deriving the Unit Quaternion to Tait-Bryan Angles conversion.

Let me start by saying I have a working solution. But I just don't understand how to get there. I've followed the well-written paper Technical Concepts Orientation, Rotation, Velocity and Acceleration,...
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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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### Determine the rotation necessary to make a line tangent to an ellipse

Question: Given the line $\ell(t) = r_0 + t \ u$ I want to rotate it about point $P_0$, such that it becomes tangent to the ellipse $(r - C)^T Q (r - C) = 1$ where $r = [x,y]^T$, $C$ is the center ...
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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 ...
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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 ...
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