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

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Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by ...
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53 views

What does the rotation group of $\mathbb{\bar{Q}}^n$ look like?

There's a structural difference between the rotation groups of $\mathbb{Q}^n$ and $\mathbb{R}^n$; in some abstract sense the former is 'small' (discrete?) while the latter is 'large'. I suspect that ...
4
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86 views

Infinite product of rotation matrices

Suppose we have a product $$\vec v=\left(\vec x^T \cdot R(\vec\varphi_1)\cdot R(\vec\varphi_2)\cdot ...\right)^T,$$ where $R(\vec\varphi_i)$ is a matrix of rotation by $3D$ angle $|\vec\varphi_i|$ ...
3
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38 views

Second derivatives of rotations

Given an exponential parameterization of a 3D rigid rotation $R\in SO(3)$ by the vector $v = (v_x, v_y, v_z)^T$ I would like to find its second derivatives at the point $v=(0,0,0)$. Using the ...
3
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29 views

Check if a point is inside a rotated 2D NACA 0012 airfoil

I've already checked the rotated rectangle problem but this is (I think!) a little more complicated. I have a CFD calculation of a 2D NACA 0012 airfoil and I need to test if a point is inside the ...
3
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48 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
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77 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
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117 views

Problems sampling from a $pdf$ over $SO\left(3\right)$

I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series: $$ f\left(\omega,\theta,\phi\right)=\sum ...
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102 views

3D rotation group

It is known that the group $\text{SO}(3)$ of rotation-matrices (matrices $A$ with det(A)=1) are generated from three parameters. This can be expressed by the fact, that any rotation matrix is a ...
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38 views

Free groups of rotations of the sphere

Is the following conjecture true: If $G$ is a group of rotations of the sphere and $G$ contains two noncommuting rotations of infinite order, then $G$ has a free subgroup of rank $2$. By the Tits ...
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22 views

Calculate new pitch and roll after rotating about the z axis

I am wanting to know how to find out the new pitch and roll values when rotating around a circle. I have become a little stuck on how to achieve this, but hopefully someone will be able to point me in ...
2
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18 views

Rotation about arbitrary point and arbitrary axis

This should be a simple problem, but I appear to not be able to get this correct. I have an object that is rotating in a circle on the $x$-$y$ plane (rotating in the $-z$ direction) at a speed ...
2
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58 views

Rotate a 3D Vector onto Another 3D Vector

I am trying to transform one triangle onto another triangle in 3D space (Right Triangles). My thought was I align the forward and left vectors, then translate the center of one to the other. ...
2
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31 views

Rate of convergence of an irrational rotation

Let $e^{i\phi}, e^{i\theta} \in \mathbb{S}^1$. Let $p(x) = x + 2k\pi$, where $k \in \mathbb{Z}$ is chosen so $p(x) \in [0, 2\pi)$. If we assume that $\phi/\pi$ is irrational, then there exists an ...
2
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284 views

Rotating a cube about an axis through opposite vertices

I have a cube made using CSS transforms that I'm trying to animate rotating about an axis going through 2 opposite vertices. What I have: Initial cube: ...
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47 views

Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
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89 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
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73 views

Find an SO(3) matrix which satisfies some linear constraints

I have the following optimization problem: $\displaystyle \min_R \sum_{i=1}^n (X_i^T R Y_i)^2$ where $R \in \text{SO}(3)$, i.e. is a 3x3 rotation matrix, and $X_i,Y_i \in \mathbb{R}^3$. If $n \le ...
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148 views

Understanding quaternions and axis angle representations

I have a sensor that gives me a quaternion. I convert the quaternion to an axis-angle representation using http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToAngle/. When I ...
2
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271 views

Desired Z axis and Yaw to ZXY Euler Angles?

I'm trying to calculate a desired pair of pitch and roll Euler angles (the XY in ZXY format) given a desired z-axis of the rotated frame (expressed in the world frame) and a specified yaw angle ...
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95 views

Help me to vizualise this falling ball on spinning Earth

The earth rotates. The ball falls in an latitude, not equator, let say in Germany. I am trying to understand how to express the ball in terms of the angular velocity on the planet. The constant ...
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244 views

Gimbal lock and zero jacobian determinant

I'm having a hard time visualizing the gimbal lock problem. Suppose $p=(a,b,c)$ is a point where the euler angle $f:\mathbb R^3\to SO_3$ has zero jacobian determinant. Then to say that $p$ is where ...
2
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359 views

4d rotations and quaternions

I have a question about 4d rotation: I programmed a little 4d game and I used the classical hyper-sphere coordinates, to rotate a vector. It works, but it has some problems :( (just for clarity I ...
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22 views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
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33 views

Transformation matrix for two rotations

If I am required to compute the full transformation matrix compromising of the following sequence of operations: rotation by $30$ degrees about $x$-axis translation by $1$, $-1$, $4$ in $x$, $y$ and ...
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32 views

rotation matrix - satellite control system

I am in charge with developping a satellie control system. There is something that I either understood incorrectly or either is wrong in the explanation. I am checking if I am able to get the values ...
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11 views

two rotations of line segment in different order

there are given are two rotations: $R_1$ and $R_2$, and a line segment $AB$. the image of $R_1R_2(AB)$ is a line segment that is parallel to the line segment $R_2R_1(AB)$. my kindly request is to the ...
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20 views

Using Levi-civita symbol to determine axis and angle of rotation matrix

One of the questions on the course involves finding the angle and axis of this rotation matrix; $$R = \gamma\ \begin{pmatrix} 0 & -2 & 1\\ 2 & 0 & 0\\ -1 & 0 & 0 ...
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13 views

Calculate the plane angle from 2D plane

I am analysing a squared plane from a perfect cube. This plane is distorted by the perspective view of a camera. I would like to know ask please, some approaches of how could I get to know the ...
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115 views

Cabri 3D - Rotating a triangle

I'm given the exercise, in Cabri 3D, to rotate the triangle T around the axis AB and lead it via the triangle To to the triangle T'. I tried to rotate the triangle T around a fixed point and then ...
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45 views

Finding volume of solid using integration.

Find the volume of the solid obtained by rotating the region bounded by the curves: $y=x^8$ and $y=1$ about the line $y=6$. I tried to use the washer method but then I cannot find two volumes that I ...
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33 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
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22 views

Is my interpretation of Rotation Matrices correct?

I've been asked to find the matrix which rotates vector $\vec{V}$ by angle $\alpha$ in the x-y plane. This I understand and I've constructed the matrix: $R_{\alpha}= \begin{bmatrix} cos\alpha & ...
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30 views

Why does $det(R) = +1$ imply right handed frame?

Let $R$ be a rotational matrix in $SO(3)$ so it satisfies $R^TR = I$ Solvng for $det(R^TR) = (det(R))^2 = 1$ yields two solutions Why does $det(R) = +1$ mean that the frame is a right handed frame? ...
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78 views

Finding eccentricity, directrix, foci of diagonal ellipse by rotating it

A problem I'm working on has the equation for a diagonal ellipse $$5x^2 + 5y^2 - 6xy - 8 = 0$$ which can be rotated 45 degrees to get the vertical ellipse $$\frac{x^2}{1}+\frac{y^2}{2^2} = 1$$ The ...
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56 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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84 views

3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
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157 views

Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship ...
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56 views

Given a corner, draw a square of known size minimizing vertex distance to a point

I've been going a little crazy trying to solve this what I think is a simple problem. Given: A known corner (~90 degrees) as defined by angle ABC. Segment AB is defined by two points on AB with ...
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75 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
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34 views

Calculate rotation on a sphere with given coordinates

I have a sphere with a fixed radius. I have a set of points on that sphere, let's say $p_1, p_2$ and $p_3$ and it's $3$D Cartesian coordinates. I rotated each of the points around the center of the ...
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36 views

Cross product uniqueness

I have following relationship between vectors $A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$ Given Data " ' " means derivative ...
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12 views

Find path between two attitudes subject to body rate constraints

Here's my problem. I have an initial orientation and angular velocity of a body and a final orientation and velocity occurring at a specified time in the future. I have control over how input ...
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42 views

Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
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43 views

Eigenvalues of rotation invariant operators on 2-sphere

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore ...
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62 views

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 ...
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34 views

Rotation of point with infinite child objects. (Chain rotation)

More of a thought experiment here, knowledge for knowledges sake. Let's say you can create infinite points that rotate smoothly and at the same speed as each other through a full revolution - let's ...
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25 views

Rotate Object relative to other Object

I have an object (cube) which has two other objects attached to it (cones). If I rotate the cube along any axis, how do I determine the rotation of the cones (which emanate at a specific angle from ...
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106 views

Finding rotation matrix from angles between plane intersections and axes

I have a proper rotation transformation between coordinate axes $\{X, Y, Z\}$ and $\{X^\prime, Y^\prime, Z^\prime\}$. What I am given are three angles, all of which have vertex at the origin: Let ...
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59 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...