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

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Non-standard 3D rotation of a set of points

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
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1answer
32 views

Geometrical calculation to determine size difference between two rectangles when rotating one

I've asked a programming question on StackOverflow here which should give you a good understanding why I'm trying to do this. I'm asking it here because it's now down entirely to the mathematics of ...
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37 views

Rotation matrix around one coordinate in N dimensions

Probably a very simple question: Given the standard Cartesian coordinate matrix, $$\begin{pmatrix}1 & \\ & 1 & \\ & & 1\\ & & & 1\\ & & & & ...
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13 views

Euler angles and composition

Suppose that I have two rotations $A, B \in SO(n)$ with Euler angles: $$A = g^{(n-1)}(\alpha^{n-1}) g^{(n-2)}(\alpha^{n-2}) \dots g^{(1)}(\alpha^{1}),\quad B = g^{(n-1)}(\beta^{n-1}) ...
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1answer
24 views

Determine similarity between two sequence of quaternions while allowing a degree of freedom around Z axis

A person holds his phone and rotates it in space in a sequence. I am able to obtain a sequence of quaternions from the phone's motion sensors representing the rotation of the phone from the phone ...
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1answer
19 views

rotate a time series with constraints for start and end

I have a time ordered series of data that I can represent in R like: ...
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0answers
13 views

How to determine roll-pitch-yaw parameters from homogeneous transformation matrix

We have a transformation matrix $T = \begin{pmatrix} cos(\theta_1) & sin(\theta_1) & 0 & l_1 cos(\theta_1) \\ sin(\theta_1) & -cos(\theta_1) & 0 & l_1 cos(\theta_1) \\ 0 ...
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3answers
38 views

what is the smallest number $n\in \mathbb N$ such that $A^n=I$? [closed]

Let $A$ be a $2\times 2$ matrix consisting of $$A = \begin{pmatrix}\sin(\pi/18)&-\sin(4\pi/9)\\ \sin(4\pi/9)& \sin (\pi/18)\end{pmatrix}$$ what is the smallest number $n\in \mathbb N$ such ...
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1answer
25 views

Vector rotation

Given a unit vector and an angle of rotation about the unit vector, how do I quickly compute the rotation matrix. I know there must be a simple formula, but I have been unable to find it.
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0answers
15 views

Apply a rotation to Euler Axis angles

I have a camera orientation in world coordinates expressed in a vector containing cameras axis angles relative to the three world $x$, $y$, and $z$ axes (as example $(0,0, 0)$ would be a camera ...
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32 views

Slope of image side for 3D rotation

I had a new idea for an experimental 3D assembler (not a rasterizer). The idea requires that I get the slope of the top, bottom, left, or right depending on the $z_n$ axis. My idea works on two ...
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16 views

Express a 90 degree rotation matrix in terms of a 180 degree rotation matrix? (both anti-clockwise)

A = [-1 0 0 ] [ 0-1 0 ] [ 0 0 1 ] B = [0 -1 0 ] [ 1 0 0 ] [ 0 0 1 ] How can i represent B in terms of A?
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15 views

Searching for a definition for n-Dimensional rotation which is cosine-distance invariant

I am wondering if it is possible to define a rotation for an $n$-Dimensional space ($n=2,3,4,5,\dots$). Given any vector $\vec v$, and knowing that it should be rotated to ...
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1answer
30 views

Yet another n-Dimensional Rotation question: is there a definition for n-Dimensional rotation which is cosine-distance invariant? (TO CLOSE)

(Moved to Searching for a definition for n-Dimensional rotation which is cosine-distance invariant, flagged it to delete it) I'm wondering if there exists a rotation definition by which the vectors ...
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3answers
40 views

How can I rotate a point 45 degrees counterclockwise around any point?

What is a formula (in terms of $x$ and $y$ coordinates) for rotating one point about another by $45$ degrees counterclockwise? I've tried using: $$x'=x\cos(-45^{\circ})-y\cos(-45^{\circ})$$ ...
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0answers
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Delaunay surfaces - plane as surface of revolution

According to Wikipedia (and other sources) "Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These ...
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34 views

Represent $90$ degree clockwise rotation about the $z$-axis as a $3\times 3$ matrix

I honestly can't find anything regarding an issue I have with transformational matrices. I understand that this matrix: $$\begin{pmatrix} \cos 90&-\sin 90&0\\ \sin 90&\cos 90&0\\ ...
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1answer
26 views

Know if a 4x4 matrix is a composition of rotations and translations (quaternions)

I am using quaternions to describe 3D transformations. A position in space is representated by a (x,y,z,1) vector, and a transformation by a 4x4 matrix, following quaternions logics as far as I could ...
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1answer
20 views

3D rotational matrix between two spherical co-ordinate systems.

So I have a classical mechanics problem where I have worked out the azimuthal and altitude angle for a vector, I then want to apply rotational matrices so that the vector is realigned with the z axis ...
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1answer
15 views

Angular momentum superposition

I am doing a space simulation. I have a spaceship and this spaceship has engines that don't push the spaceship through its center of gravity. These engines can therefore give the spaceship angular ...
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0answers
21 views

How to calculate end point of vector using quaternion?

How can the end-points of the three orthonormal vectors representing local coordinate system of point (p.x, p.y, p.z) be calculated given rotation represented by quaternion in global coordinate system ...
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2answers
35 views

Rotation Equivalence using Quaternions

I'm given a statement to prove: A rotation of π/2 around the z-axis, followed by a rotation of π/2 around the x-axis = A rotation of 2π/3 around (1,1,1) Where z-axis is the unit vector (0,0,1) ...
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1answer
34 views

Obscure Rotation Matrix

The points of the shape after the shear are $(-0.25, 1)$, $(1.75, 1)$, $(0.25, -1)$, $(-1.75, -1)$. Other than that the only other information given is that the vertical edges of the original shaded ...
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1answer
32 views

How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
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1answer
17 views

Rotate a vector into a plane spanned by two other vectors

In an application test that I had to do for a job recently, I was asked the following question (I quote): “Given three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. Compute the rotation ...
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2answers
55 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - ...
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1answer
43 views

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?
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1answer
27 views

Two body problem (rotation around a fixed central point)

Is there a way which isn't physics related, but just using pure maths to find the solution to the following problem: If i have two lines of different lengths at t=0 overlapping each other. They are ...
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0answers
6 views

Can gimbal lock be avoided with Euler angles by resetting the axes?

When using Euler angles to describe rotations of an object can we avoid gimbal lock by simply resetting the axes after each rotation? Alternatively when we conduct rotations can be not just ensure ...
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0answers
10 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
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9 views

Confusion about coordinate transforms

Lets say I have a camera aligned with the world coordinates system. I rotate it by 180 degrees around the z axis and then by 20 degrees around its new y axis. I have been reading about Euler angles ...
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1answer
18 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...
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2answers
21 views

Find center of rotation after object rotated by known angle (2D)

I need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle. Previously I would simply find the center of the object, rotate 180 ...
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0answers
31 views

Linear transformation with clockwise rotation on z axis

Let $T$ be a linear Transformation from $\mathbb{R}^3$ to itself such that $T$ is $60^{\circ}$ clockwise rotation with fixed z-axis (i.e, rotate the space according to the z-axis) where $\mathbb{R}^3$ ...
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1answer
4 views

Find Z Rotation based on X and Y Vector

I have a vector $(x,y) = (x_2 - x_1, y_2 - y_1)$. I have an arrow pointing to 0 degrees. With vector $(x, y)$, how can I find the number of degrees (0 - 360) that will be the direction the arrow ...
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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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1answer
79 views

Is it true that a arbitrary 3D rotation can be composed with two rotations constrained to have their axes in the same plane?

I am interested in decomposing an arbitrary rotation in 3D space into the product of two rotations which are constrained to have their axes in the same plane (for instance x-y plane). Statement of ...
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12 views

Understanding Check: 3D Rotational forces as a vector

Am I correct in understanding that the magnitude of rotational forces in 3 dimensions can be expressed as a vector, from the origin, whose x, y, and z components represent the magnitude of the ...
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0answers
19 views

Rotation in spherical coordinates (w/o Cartesian)

What are the rotation matrices in polar coordinates? Which matrices I should multiply by a unit vector $(1, \theta, \phi)$ to rotate the latter around basic axes to angle $\beta$? Used notation: ...
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1answer
40 views

Rotation matrix in spherical coordinates

When given arbitrary point on a unit sphere $a = (\theta, \phi)$ and an arbitrary axis $\vec{A}=(\Theta, \Phi)$, can we have an algebraic expression for $a_1=(\theta_1, \phi_1)$ which is a rotation of ...
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14 views

Unexpected Asymmetry in Tate-Bryan angles extracted from perturbed Quaternion

I’ve checked some references and the following MATLAB code seems to be correct for converting a quaternion to body roll, pitch, and yaw Tait-Bryan angles respectively. ...
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19 views

How to calculate translation matrix?

I have a point cloud, which consists of three points. First point cloud has points A(xa, ya, za), B(xb, yb, zb) and ...
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1answer
19 views

What does rotation northwards and rotation eastwards mean?

The following quote is causing me trouble: " For instance, suppose we start off at ($0^\circ$N, $0^\circ$W), which is just off the Atlantic coast of equatorial Africa, and rotate $90^\circ$ ...
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2answers
25 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
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Position Vector - Using Geometry

Given Data and Specifications in the problem $O_i,O_j$ represent two co-ordinate systems with base orthogonal frame vectors $x_j,y_j,z_j$ and another frame $x_i,y_i,z_i$ with respect to that ...
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1answer
76 views

Eigenvectors of a Rotation Matrix

The eigenvector of the rotation matrix corresponding to eigenvalue 1 is the axis of rotation. The remaining eigenvalues are complex conjugates of each other and so are the corresponding eigenvectors. ...
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1answer
34 views

Calculus: Volume by rotating curve

R is the region in the first quadrant that is bounded on the left by the y-axis, on the right by the curve $x = \tan(y)$, and above by the line $y = \pi/4$; l is the line $x = 1$. I came up with the ...
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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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Rotation in configuration space.

Let $R_\psi$ be the rotation in configuration space around a vector $\bf{e}_\psi$ for an angle $\psi$. How is that the space rotation in configuration space have: ...
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1answer
18 views

Given unit quaternions $q_0,q_1$, find $q$ such that $q_1 = q^* q_0 q$

I rotate an object in space and find two orientation (unit) quaternions. $q_0 = {}^{M_2}_{M_1} q$ is the orientation at the 2nd position relative to the 1st position, measured in frame M. $q_1 = ...