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

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How do I rotate a rectangle of latitude and longitude?

I have a rectangle with its corners specified in latitude and longitude. I would like to rotate it about it's centre a certain number of degrees. I was using longitude as an x value and latitude as a ...
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1answer
21 views

Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations?

I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which ...
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1answer
44 views

Rotated arc in $\mathbb{R}$ [on hold]

We have got the arc $$A = \{(x,y) \in \mathbb{R} \mid x^2 + y ^2 = R ^2, 0 \leq x \leq R, 0 \leq y \leq R\}$$ and $R$ is positive real number. What is the area of ​​rotational figure obtained if ...
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Abstract question about the rotation in 3D

I've made the following observersations: Zero-dimmensional entities, points, don't really rotate. One-dimmensional entities, lines, rotate about a stationary point. Two-dimmensional entities, ...
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19 views

Stopping angular momentum to obtain a particular angle

While the overall project relates to software development, it boils down to a simple (i think) physics problem. I have a joint (a motor, pretty much.) that needs to move to a specific angle. I can ...
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1answer
40 views

Calculus question that is driving me nuts. Rotation volume. [closed]

Find the volume formed by rotating the region enclosed by: $y=\sqrt[5]{x}$ and $y=x$ about the line $y=25$.
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1answer
57 views

Basic Eigenvalue Question

The rotation matrix $$T=\left[\begin{array}{c c}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right]$$ has no eigenvectors as an operator $T:\mathbb{R}^2\to\mathbb{R}^2$. Here ...
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1answer
32 views

Rotation matrix of triangle in 3D

How can I find out the rotation matrix of a right angle triangle defined by 3 points in 3D space (assuming the un-rotated triangle faces the x axis)
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Find the angle of rotation about a vector caused by application of a rotation matrix

I have a rotation matrix $R$ and a unit vector $\mathbf{v}$. How can I find the angle of rotation about $\mathbf{v}$ caused by the application of $R$?
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1answer
56 views

Rotations - linear or quadratic?

In linear algebra rotations are represented by matrices, i.e. linear transformations How do you formally prove that rotation is a linear transformation? But this page is very interesting ...
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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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1answer
30 views

Rotating a point on a circle

The wheels on a bicycle have $r$-inch radii. After the front wheel picks up a tack, the bike rolls for another $d$ feet and stops. How far above the ground is the tack? I've been thinking about this ...
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equation of transformation reference frame

I want to determine the equation of transformation from geographic to Eulerian latitude, if my reference frame is obtained rotating an Euler pole to the North pole.
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17 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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1answer
32 views

What's the difference between these rotations?

1) Each point on the coordinate plane is rotated $\theta$ degrees about the origin. 2) Each point $P$ with the coordinates $(x,y)$ is rotated $\frac{\pi}{4}$ radians about the origin. The answer ...
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0answers
18 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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Computing the coordinates of a point, offset from a rotated point.

Good day. I have a question which should be easy but I have not been able to figure it out. The coordinates of a point on a unit circle, given an angle, is $$\begin{align} x &= \cos(\alpha) \\ y ...
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1answer
12 views

Change of Eigenvalues of Ellipsoid Tensor during Rotation

I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix ...
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Finding the smallest possible set of euler angles, without changing the end-rotation

Euler angles aren't unique, so one rotation can be represented through different combinations of pitch, yaw and roll. In my case I have a set of euler angles and I need to find out if there's a ...
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3answers
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What kind of transformation an upper triangular matrix represents

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal ...
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0answers
23 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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25 views

Vector Magnitude during rotation

Probably something I should now already but this is confusing me no end! Lets say we have a force which is directed at 69 degrees inclination (from the X axis) with a magnitude of 500, shown below: ...
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2answers
46 views

What's the intuition behind the 2D rotation matrix?

Can anyone offer an intuitive proof of why the 2D rotation matrix works? http://en.wikipedia.org/wiki/Rotation_matrix I've tried to derive it using polar coordinates to no avail.
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Cryptology - Compare the amount of work the cryptanalyst is likely to require - Single vs. Double rotation

"Suppose a cryptanalyst suspects that SECEC SYHRI IRFET SSETE INLST AFNIA FSOAI HFSRT TEATE was obtained by a succession of two rotations with different block lengths and rotation amounts. Compare ...
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0answers
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Transforming euler rotations to other coordinate system

I have a global $3D$ coordinate system, and it's transform matrix is an identity matrix. I also have an object with it's own, local coordinate system and it's transform is not identity matrix. Now I ...
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1answer
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Estimate Rotation and Translation from two sets of points in different coordinate systems

I got one set of 3d $(x,y,z)$ points $( \# \geq3 )$ located in two diffent coordinatesystems. Is it possible to estimate the rotation and translation between these systems? Something like $$ ...
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Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
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1answer
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What are the coordinates of a point on a rigid body after a rotation in 3D Euclidean space, given the initial coordinates and a center of rotation

Main question Let ($x_p$, $y_p$, $z_p$) be the initial coordinates of a point $P$ on a rigid body in a right-handed 3D Euclidean space. Let ($x_r$, $y_r$, $z_r$) be the coordinates of a center of ...
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20 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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2answers
33 views

“Simple” math question about length and rotation relations

I'm currently building a robot arm as a hobby, and I'm still in the planning phase. But I've encountered a small problem, where my knowledge doesn't suffice. This is what I am trying to achieve: I ...
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0answers
35 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 ...
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1answer
22 views

Given an axis of rotation and an angle, work out the rotation angles around x,y,z axis

I want to convert from one 3D rotation convention to another. The first convention has an axis of rotation, $\boldsymbol{r}$ and an angle $\theta_r$ to rotate about this axis. The second convention ...
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2answers
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A question about the rotation number of homeomorphisms of the circle

Let $f: S^1 \rightarrow S^1$ be an orientation-preserving homeomorphism of the circle and let $F: \mathbb{R} \rightarrow \mathbb{R}$ be any lift of $f$. Usually one defines the rotation number ...
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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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1answer
26 views

Euler angles for mapping three points on a sphere to three other

Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be points on the unitary sphere, so that $\|\mathbf{a}\| = \| \mathbf{b} \| = \| \mathbf{c} \| = 1$. Let $\mathbf{a'}$, $\mathbf{b'}$, $\mathbf{c'}$ be ...
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1answer
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Analytic geometry - rotation + translation

In $K=O\vec{e_1}\vec{e_2}\vec{e_3}$ I have to find the analytical representation of the screw motion( rotation + translation) $\psi$ with a rotation axis $g$ given by the points $A(5,-4,3)$ and ...
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1answer
12 views

What is the the generalization of Euler angles for O(3)?

Is there a generalization of Euler angles that handles the case where inversions are allowed? I'm trying to figure out how to parameterize elements of $O(3)$ for a computer application; if I was only ...
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31 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 ...
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3answers
58 views

Does R' = R*R'*R^(-1) hold?

Given two 3D rotation matrices ($3\times 3$) $R$ and $R'$, does this equivalence hold?: $R' = R*R'*R^{-1}$ My intuition tells me so, but I can't find a formal proof for it. Thanks.
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2answers
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Rotation between two vectors as a function of time (one parameter rational motion design)

Given a time varying vector: $\mathbf{w}(t) = \mathbf{u} + t\mathbf{v}$ I would like to find a rotation matrix $\mathbf{R}(t)$ that rotates the positive x-axis $[1, 0, 0]^T$ onto the vector ...
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2answers
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Set up: Volume by Integration

Having difficulty setting up the equation. Find Volume bound by $$y = x; y = 0; x = 2; x = 4$$ and rotated about $x = 1$ The issue I'm having is that I get a final answer of $\frac{16\pi}{3}$ ... ...
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20 views

Rotation of a Polygon Direction

Whenever I look at a problem that involves rotating a geometric figure with given vertices a number of degrees, how come it should be rotated counterclockwise, unless otherwise stated?
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What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + ...
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37 views

How to obtain relative rotation?

I have two rotations, each of which can be described as a roll, pitch, and yaw (in radians): $$ r_1 = (3.14159, 1.57080, 1.6) $$ $$ r_2 = (3.14159, 1.57080, 1.4) $$ I am interested in the relative ...
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1answer
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proof for euler-rodrigues formula - matrix form

I need for a matrix representation. Exactly I want to know how to get the Euler-Rodrigues formula in a matrix form like here. Thanks!
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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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1answer
29 views

Find volume of revolution

Find the volume: $$x=y^2; x=1-y^2; \text{ rotated around }x=3$$ Use the disk/washer method. My inner radius is $3+y^2$ and outer radius is $3-y^2$ and the limits of integration are $\frac1{\sqrt2}$ ...
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composition of rotation matrices

We wish to construct a general rotation $\mathbf{R}$ of a coordinate system by composing three elementary rotations $\mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_3$, so that a vector $\mathbf{v}$ is rotated ...
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1answer
36 views

How can I rotate a coordinate around a circle?

I have been searching the answer for a few weeks now, but I cannot find it, and I decided asking here was worth a shot. I need to rotate an object in a circle around a central point. All I know is ...
2
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1answer
27 views

Eigenvectors of this matrix - what's the relation to rotation operator?

I have found the eigenvalues to be 0, 1 and 2. The corresponding eigenvectors are: $\frac{1}{\sqrt 2} (1 , -1, 0)$ and $(0, 0, 1)$ and $\frac{1}{\sqrt 2}(1, 1, 0)$. I found that when $x^2 + 2xy + ...