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Questions tagged [rotations]

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

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How/why does the ICP algorithm using quaternions work?

I've come across the Iterative Closest Point algorithm using quaternions (as described in "A Method for Registration of 3-D Shapes" by Besl and McKay) and I'm wondering, why it works. To me it seems ...
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4answers
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3D rotation orientation

My problem involves finding a formula for this: You're given a line that passes through the origin in 3D space (more specifically, a point on the line). You are then given a point and an angle. The ...
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1answer
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calculating new 3D position on sphere with angular velocity vector

I feel like this is actually pretty simple but still could not find any solutions so far... I'm trying to calculate the movement of a point in a rigid rod with the equation $ \dot P = [ v + \omega \...
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Convert spherical coordinate rotation to aircraft-axes rotation

Spherical coordinate system (I'm using common mathematics convention) Aircraft axes system To be even more specific, the aircraft-based system that I'm working on has these properties: 1. It's ...
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2answers
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Rotation of $e_1 \in \mathbb{R}^n$ in angles along the axis

I have the vector $e_1=(1,0,...,0)^T$ in $\mathbb{R}^n$. I would like to rotate it by angle $\theta_2$ along axis $x_2$, resulting in the vector $r_1 = (\cos(\theta_2),\sin(\theta_2),0,...,0)^T$. ...
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Convert velocity vector to yaw roll pitch Tait Bryan

I have a cartesian position and velocity vector describing the flight path of an object in the format "time posX posY posZ velX velY velZ" and want to convert it to a "time posX posY posZ ang1 ang2 ...
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Align PCA axis to XYZ axis [closed]

I have a 3d point cloud. Before proceed a PCA in the data, I'd like to align the eigenvectors found on the process to the XYZ-axis. How can I do that task? @EDIT: The Principal Component Analysis ...
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Uniqueness of a vector given two rotations

My question concerns whether it is possible to determine a unique vector given two of the vectors rotations. Assuming we have three known vectors in 3D; $a$, $b$ and $c$, and two angles $\phi$ and $\...
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2answers
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If two rotation matrices commute, do their infinitesimal generators commute too?

Suppose that $e^A$ and $e^B$ are two rotations in $\mathrm{SO}(n)$. If $e^{A}e^{B} = e^{B}e^{A}$, can we conclude that $e^{A+B}=e^Ae^B$? More importantly, can we say that $AB=BA$? I'm particularly ...
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How to know when a 360 rotation was performed around any given axis?

If I have any given axis, e.g. $\frac{1}{\sqrt{2}}\left[ \begin{array}{ccc} 0 & -1 & 1 \end{array} \right]$ and a rotational speed of $\omega = 1.5$ [deg/s] around that axis. How can I check ...
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3answers
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Linear operator and rotation matrices

I have come across a question involving a linear operator $A$ that is represented by the following matrix: $$ \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix} $$...
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1answer
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Confused about rotation matrices

Applying a rotation matrix to a vector means shifting its coordinates to perform the rotation effect. Applying a rotation matrix to a model at the origin $(0,0,0)$ is not the same at performing a ...
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4answers
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The difference between applying a rotation matrix to a vector (points) and to a matrix (transformation)

Suppose that the rotation matrix is defined as $\mathbf{R}$. Then in order to rotate a vector and a matrix, the following expressions are, respectively, used $\mathbf{u'}=\mathbf{R} \mathbf{u}$ and ...
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Random projection of a fixed point

In the book "High-dimensional probability by Vershynin", page 111, in the proof of Johnson-Lindenstrauss Lemma, let $E$ be a random $m$-dimensional subspace in $\mathbb{R}^n$ uniformly distributed in ...
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1answer
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Which direction is clockwise when rotating around x-axis in 3D?

Picture 1 shows a demonstration that rotations around an axis is positive for clockwise directions. An example later on, picture 2, applies a rotation matrix for 60 degrees in the x-axis for a ...
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1answer
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Rotating tetragon/square with matrix algebra

So I need help with my home assignment. I would have to rotate a tetragon from one point. Tetragon is given with letters ABCD. A=(3;1), B=(7;3), C=(2;6) and D=(0;2). I have to rotate the tetragon 70 ...
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Does this machine independent to the order of rotation?

I've been tried to figure out that the order of rotation (not the vertical bar in the picture) matters in the axes described picture below. As described in the picture, 1. Tilt does not affect the ...
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4answers
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Rotation of axes by 45 degrees

I was reading a book in which it is mentioned that: Rotate coordinate axes by $45$ degrees so that a point $(x,y)$ becomes $(x+y,y-x)$ . Here is image 1 Here is image 2 I can't understand how the ...
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1answer
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Understanding the cosine as a partial derivative.

From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis): To go on to three and, later, four dimensions, we find it convenient to use a more compact ...
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1answer
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Tail of rotated vector not properly defined using Rodrigues' rotation formula

I programmed https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula for some graphics stuff, but encountered what seems to be a math problem with the formula, as discussed below... The wikipedia ...
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0answers
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Calculate roll from yaw and pitch?

I came across an issue earlier today where I have created a look-at camera in 3D space, and applied yaw and pitch to it to rotate around the target properly. The problem is that introducing local ...
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Search for projection on a special matrix space with regard to Frobenius norm(computer vision background)

Background Define essential space as $$\varepsilon=\{E \in \mathbb R^{3\times3}|E=\hat{T}R\}$$ $$\hat{T}\in\{S\in \mathbb R^{3\times3}|S^T=-S\}$$ $$R\in\{A\in\mathbb R^{3\times3}|A^TA=I,\det(A)=1\}$$...
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2answers
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Proofing the derivative of Rodigruez's Rotation Formula equals the formula for relating the linear velocity of a point to the angular velocity.

I want to proof that the derivative of Rodriguez's Rotation Formula equals the formula for relating the linear velocity of a point on a moving body to the angular velocity (following the footsteps of ...
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0answers
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What is the role of representation theory?

This question is a bit vague mostly because of my very limited knowledge of group theory. But I would like to know if there is some intuitive example of representation theory that gives a layman an 'a-...
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0answers
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From direction cosine matrix to Cardan angles - Two different Matlab codes, different results

I have two Matlab codes to get the Cardan angles (ZXY sequence) starting from a given direction cosine matrix (dcm), but the results are different. Why? The dcm is: ...
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1answer
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Finding new position of a point in $\Bbb R^2$ after rotating it about a vector in $\Bbb R^2$.

Suppose $(x,y) \in \Bbb R^2$. Suppose we rotate this point about a vector $(a,b) \in \Bbb R^2$ through an angle $\theta$. Find the coordinate of the new point. What I have done is as follows $:$ ...
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1answer
42 views

Difference in rotation matrix

I have two objects A and B, with a start transformation matrices MA1 and MB1 (they include translation, rotation and scale). End matrix of the first object is MA2. How do I apply the same rotation as ...
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0answers
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Id $f,g$ orientation preserving cricle-diffeomorphisms, then $\rho(g^{-1}\circ f \circ g) = \rho(f)$.

Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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1answer
33 views

Universal formula to calculate rotating by angle

I am generating roads and buildings that belong to them and since I want the streets to be rotated and then connected with each other, I need to rotate both them and their respective buildings. Is ...
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Calculating a transformation matrix for a given pair of matrices

Simple deduction tells me that given square matrices $A*B=C$, we can calculate $B=C*A^{-1}$, but writing the code using OpenCV's cv::Mats things don't calculate ...
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1answer
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Rotate isometric projection to square projection

UPDATE 2: I got it working now and i wanted to post my solution for future reference, if someone has the same question then i had. First a screenshot of the final rendering: final rendering The ...
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3answers
142 views

Question about rotation $2\times 2$ rotation matrices

How do I prove that, if for a $2 \times 2$ matrix $A$ and a fixed integer $n> 0$ we have that $$A^n= \begin{bmatrix} \cos{x}& -\sin{x}\\ \sin{x} & \cos{x}\end{bmatrix} ,$$ for some real $x$,...
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2answers
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Standard matrix of a rotation on a vector in $\mathbb{R}^3$?

$\begin{pmatrix}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{pmatrix}$ defines a rotation of any angle on a vector in $\mathbb{R}^2$, but how does it work in $\mathbb{R}^3$ ? (I'...
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1answer
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Rotating 3D points around a Z axis

I'm messing around with a script for the game and I'm trying to rotate a house. House consists of multiple objects. I need to rotate a house for certain degrees but I'm not very good at math and ...
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Distributive property of a matrix on a cross product

Let $\vec x, \vec y \in \mathbb{R}^3$ and $\bf A $ be a $3 \times 3$ real matrix. Under what conditions does $\bf A$ distribute over a cross product: $$ \mathbf{A} (\vec x \times \vec y) = (\mathbf{A}...
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Equality of Lie-Cartan coordinates of the first kind and Lie-Cartan coordinates of the second kind

For a set of basis $(w_1,w_2,w_3) \text{ of linear space } \mathbb R^3$, define Lie-Cartan coordinates of the first kind: $$ R_1 = \exp(\alpha_1 \hat{w_1} + \alpha_2 \hat{w_2}+\alpha_3 \hat{w_3}). $...
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0answers
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Products of subgroups of Euclidean Group

So, I'm reading a book and I need help with some stuff. The book defines product of subgroups $G_1, G_2$ of $G = E(n)$ as $G_1G_2 = \{g_1g_2|g_1 \in G_1, g_2 \in G_2\}$, which is not necessarily a ...
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1answer
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Sum of vectors and the effect of applying rotation and scale factors

I have a set of vectors $\mathbf{V} = [\mathbf{v}_{1} \cdots \mathbf{v}_{M}] \in \mathbb{R}^{N \times M}$. The sum of these vectors will form another vector $\mathbf{w} \in \mathbb{R}^{N}$, as is ...
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2answers
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Why doesn't a 20 degree rotation change the slopes of $y=x$ and $y=\frac{x}{2}$ by the same amount?

It seems that if I rotate different lines (lying in the same quadrant) the same number of degrees they move different amounts (in terms of their slope). (where the rotation is such that all the lines ...
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0answers
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Finding axis of symmetry given n points and their surface normals

Given a roughly torus-shaped 3D object: it has cylindrical symmetry, one axis of rotation, and is symmetric with respect to any angle around that axis. Imagine a donut, but with any number of ridges ...
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35 views

Quaternion that transforms a point like a 2D angle

Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem. The algorithm I've implemented in 2D works as follows: Given the points $A (A_x, A_y)$ and $B (B_x, ...
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Two rotations of different axes

If we know that any rotation of $SO(3)$ can be disturbed as a product of two specular symmetries and that one of them can be arbitrarily chosen between the planes that contain the axis of the rotation....
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What is the distribution of the unit sphere multiplied by a uniformly distributed scalar?

We know that any rotationally-invariant distribution can be written as X = RU where R is |X|, U ~ Unif[S^(n-1)], and R is independent of U. In words, we can choose a direction vector using the unit ...
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Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions: (1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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Using 6th dimensional vector to rotate a tesseract

I'm trying to rotate a tesseract in 4D space for a project. This shows I can use bivectors to "to generate rotations in four dimensions." Following exterior algebra and finding the wedge product, I'...
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1answer
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Proof of Logarithm map formulae from $SO(3)$ to $\mathfrak {so}(3)$

According to exponential map, there also exist a logarithm map $$\log:SO(3) \to \mathfrak {so}(3).$$ Suppose a vector $t \in\mathfrak {so}(3)$ and $t=\|t\|w$, according to exponential map $$R = \cos\|...
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2answers
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Is there a technique to derive the groups of rotations of various objects?

Apart from simply memorising them or being able to visualise them on the spot and jot them down is their any way to derive the group of rotations for various shapes in 3-d. (2-d is easy enough to just ...
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motion of a rigid cube

A rigid cube is in motion. At the time depicted in the figure the face $ABCD$ is vertical, the velocity of vertex $A$ is vertical down with value $v$, the velocity of vertex $C$ is vertical up with ...
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2:1 Diametric rotation matrix for a 2D orthographic projection

I asked this question in the game dev stack exchange, but didn't get a response. This is more of a math question with an application in game development so I hope it's alright if I ask here. I'm ...
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1answer
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How to block diagonalize a real skew-symmetric matrix of 3*3

Suppose $t = [t_1,t_2,t_3]^T\in \mathbb R^3,t \neq 0$. Then define $$t^{\land} = \begin{bmatrix} 0 & -t_3 & t_2 \\ t_3 & 0 & -t_1\\ ...