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

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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 ...
5
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154 views

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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36 views

Get bounding rectangle segments of a rotated rectangle (matrix?)

My problem: I have: $x$, $y$ & $\alpha$ - the aspect ratio $o$:$p$ (red rectangle) I want to have $n$ & $m$ in dependancy of $x, y, \alpha, o, p$ I tried to figure it out with ...
4
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0answers
99 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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0answers
44 views

Why don't more celestial bodies exhibit higher-order rotations?

It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have ...
3
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0answers
34 views

Why doesn't the Sine or Cosine rule work when I'm resolving moments about P in this question?

I originally posted this on Physics stack exchange, but some users said it was more suited here, although the moderator who reviewed it decided not to migrate it, as it has some physics jargon- but I ...
3
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0answers
49 views

Davenport's Q-method (Finding an orientation matching a set of point samples)

I have an initial set of 3D positions that form a shape. After letting them move independently, my goal is to find the best rotation of the original configuration to try to match the current state. ...
3
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33 views

Rotors/Quaternions: double reflection question

I am trying to learn/understand quaternion. I found this reference (among many others): http://www.geometricalgebra.net/quaternions.html It states (see attached screenshot of that page), that to ...
3
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28 views

By or through for a rotation

"Rotated through pi rad" vs "Rotated by pi rad" I have heard both used and also heard mentioned that there was a mathematical difference between the two. Is this true or can they be used ...
3
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0answers
60 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
3
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239 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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57 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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0answers
100 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 ...
3
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80 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 ...
3
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126 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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0answers
13 views

Show that $δ_{KL}$ is a Cartesian tensor

By using the definition of the Kronecker delta $δ_{KL}$, show that $δ_{KL}$ is a Cartesian tensor, that is $δ'_{MN} = L_{MK}L_{NL}δ_{KL}$ under the rotation $X_K = L_{MK}X'_M$. Solution: Using the ...
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25 views

Alternative to affine space

I've been reading up on affine geometry. An affine space (correct me if I'm wrong) is a set of "points" along with a set of translations on those points such that for any two points $P, Q$ there ...
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146 views

Wind vector transformation from Gaussian grid to displaced pole grid

I have been given the "u" and "v" component with respect to an earth coordinate reference system(Gaussian grid - https://en.wikipedia.org/wiki/Gaussian_grid ...
2
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0answers
189 views

Rotating a 3d-ellipse equation?

So I have very limited Linear Algebra knowledge, and I'm trying to program a computer graphics application in Android using OpenGL. I understand my design is not great, so if you have questions as to ...
2
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0answers
40 views

Identity component of SO(2,1)

I am working on Lie groups, and I have several difficulties to show that the identity component of SO(2,1) is the product of an euclidian rotation fixing a vector X and an hyperbolic rotation in a ...
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836 views

Decompose 3D rotation matrix into rotation around x, y and z-axis

I have a rotation matrix R, that produces an arbitrary rotation in a 3D space. I would like to decompose it into 3 rotation matrices Rx, Ry and Rz so I can use and apply only xy in plane rotation ...
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72 views

How to use 2D Translation and Rotational error to get offset value for new point?

Here I am trying to detect FIDUCIAL points on PCB in real time using camera. After googling for Two days and reading many post and blog. I found that I have to do something called translational error ...
2
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0answers
53 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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145 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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0answers
90 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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143 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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0answers
127 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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0answers
42 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 ...
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0answers
76 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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634 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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0answers
74 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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476 views

Bivariate normal distribution; rotation; diagonal covariance matrix

Let $Z\sim N(0,\Sigma)$ with $$ \Sigma=\begin{pmatrix}\sigma_1^2 & p\sigma_1\sigma_2\\p\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix} $$ whereat $\sigma_i^2=\text{var}(Z_i), ...
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0answers
139 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} ...
2
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0answers
83 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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0answers
180 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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0answers
390 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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0answers
102 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 ...
2
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0answers
264 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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0answers
410 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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0answers
25 views

Notation: rotation matrix with a condition

I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused: Is the top part ...
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0answers
28 views

Projection from high dimension to lower, for visualization

I want to project high dimensional data points onto 2D screen coordinates, for visualization purposes. I want to be able to control the angles of projection manually (eg, with the mouse). I have ...
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0answers
23 views

Vector on a sphere

I have for some time tried to understand the math behind explained in this post, but seem to not grasp. I think the way i visualize it might be incorrect, which make harder for me to grasp what is ...
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0answers
30 views

Unifying Transformations in Complex 3-Space

I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the ...
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0answers
15 views

Clarifying the term “rotation order” for combined rotations

I'm finding different answers when it comes to combine a set of rotations (matrices or quaternions). Let's assume we want to combine the rotations Ra, Rb and Rc in the following way: rotate (an ...
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0answers
30 views

Matrix of rotation

I am haunted by a question. Consider a vector $v=\begin{bmatrix} a\\ b \\ c \end{bmatrix}$ is firstly multiplied by $R_1= \begin{bmatrix} \cos(\theta_1) &-\sin(\theta_1)&0\\ \sin(\theta_1) ...
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0answers
8 views

To prove that a generator-candidate is sufficient to find all elements in $SO(3)$

I am attempting to prove that some sequential series of rotation axes $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\in\mathbb{R}^3$ is enough to generate all possible rotations when making a full ...
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0answers
29 views

Normalizing disturbed rotation matrix

I was playing with simulations of Euler's equations of rotation in this question. This involves integrating an ordinary differential equation of a rotation matrix, $R$, which is calculated for all of ...
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0answers
33 views

How to integrate 3D rotations (orientations)?

What is the 3D rotation equivalent of integrating (or a simpler version of the problem, simply evaluating or enumerating) all the values? For example in one dimension we have the possibility of an ...
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0answers
36 views

rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3) $) of this surface and it originates from the centre of the ...
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43 views

Group of rotations of rational multiples of $\pi$

Consider the euclidean plane $\mathbb{R}^2$ and denote the rotation around the origin with angle $\theta$ by $$R_\theta:\mathbb{R}^2\rightarrow \mathbb{R}^2$$ Now consider a fixed angle $\theta_0\in ...