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

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Vorticity of a rigid body.

Consider a fluid in solid body roation about the z-axis with angular speed $\varOmega$ Derive an expression for the velocity field (u(x,y), v(x,y)) and show the vorticity field is the same at every ...
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1answer
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Translation or rotation? Identify $R_{C,-120} \circ R_{B,-60} \circ R_{A,-180}$

Let $ABC$ be a right triangle that is oriented clockwise and has angles of $90, 30, 60$ at vertices $A,B,C$. Identify $R_{C,-120} \circ R_{B,-60} \circ R_{A,-180}$ I started out with: $R_{B,-60} ...
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2answers
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Rotations/Transformations with Complex Numbers/Eulers Formula

Hello, I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral ...
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Vorticity field, rigid body

Derive an expression for the velocity field (u(x,y), v(x,y)) and show the vorticity field is the same at every point (x,y)? . When refering to the velocity field , is it refering to the components ...
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3answers
56 views

Clockwise rotation of $3\times3$ matrix?

I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me. Find the $3\times3$ matrix which represents a rotation clockwise through ...
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Scaling in the parameter of the rotation matrix

For the distance function $(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2$, the rotation matrix is $R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta ...
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1answer
8 views

Rotation matrix and invariance of norm squared

I was wondering how the distance function $(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2$ can be shown to be invariant under the rotation matrix $ \begin{pmatrix} cos\ \theta & - sin\ \theta ...
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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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1answer
20 views

Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ ...
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1answer
30 views

Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
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1answer
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Correspondence between rotations and pairs of antipodal unit quaternions

I'm having some trouble understanding how rotations of $\mathbb{R}^3$ correspond to antipodal pairs of unit quaternions. In section 1.5 of his Naive Lie Theory, John Stillwell proves the theorem that ...
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1answer
14 views

Identifying translations and rotations as compositions.

I am having trouble understanding the below which are the ones underline in red and blue. For the red: Why is that $R_{A,90}(A)=A$ and that $\tau_{AB}(A)=B$ As for the blue: Why is that ...
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0answers
13 views

What is or how do you get the rotational matrix of 4-D vector onto the xyz-space?

which would make the 4-D component 0. To be honest I'm not really sure how 4-D rotations work. I know about the simple rotations but not the mechanism in how it rotates, and I'm not sure whether to ...
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1answer
16 views

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
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1answer
10 views

How to get a Right Triangle's points' coordination in the space?

I have a Right Triangle with equal legs of 1 unit long rotated on 3 individual angles in the space like in the picture below: As could be seen in the picture, the input I have are the angles 'a' ...
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19 views

Inverse of a rigid transformation

I would be grateful for any help with the steps required to complete this calculation. You may assume that I have some experience with matrices from before, but I am obviously no master! So we have ...
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37 views

How to draw regular tetrahedron from center?

How to calculate all four points of regular tetrahedron if you have x,y and z for center point and x, y and z axis rotation and size of tetrahedron? I want to write this in java script and this is ...
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Let A,B,C be the vertices of the triangle, find the center of the following rotations

Let A,B,C be the vertices of the triangle, find the center of the following rotations: a) $R_{A,\frac{\pi}{2}} \circ R_{B,\frac{\pi}{2}}$ Two rotations that are composed together is another ...
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1answer
22 views

Identifying compositions of reflections, and rotations in a hexagon

Let $ABCDEF$ be a regular hexagon that is oriented clockwise (so that a rotation from $A$ to $B$ to $C$ to $D$ to $E$ to $F$ is clockwise). i) Identify $R_{D,120} \circ R_{A,60}$ which are two ...
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Intersection of the composition of two glide reflections

i am taking a geometry course and we are learning about isometries. I am having a hard time with glide reflections and this problem is giving me some issue, mainly because my professor usually tells ...
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2answers
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How to translate a vector and then rotate by a point

I am trying to do this problem: Identify the combination formed by first translating by the vector $(2,0)$ and then rotating by $90$ degrees about $(0,0)$. but I'm a bit confused so, I ...
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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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4 D rotation matrix

I have 2 vectors with 4 elements each and they are perpendicular to each other. z = [ -0.0310 -0.0894 -0.9451 -0.3128] and w = [0.9451 0.3128 -0.0310 -0.0894] how do i compute the 4x4 rotation ...
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2answers
37 views

Find the coordinates of center for the composition of two rotations

The combination of a clockwise rotation about $(0, 0)$ by $120◦$ followed by a clockwise rotation about $(4, 0)$ by $60◦$ is a rotation. Find the coordinates of its center and its angle of rotation. ...
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2answers
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Visualizing why rotations preserve orientation

It's clear geometrically that if you have two vectors in $\mathbb{R}^3$ a rotation will preserve their lengths and the angle between them. But how do you visualize that a rotation preserves ...
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2answers
37 views

Is rotation in $\mathbb{R}^d$ unique?

Let $\boldsymbol{u} \in \mathbb{R}^d$ such that $||\boldsymbol{u}||_2 = 1$ be a directional vector. Let $Q_{\boldsymbol{u}} \in \mathbb{R}^{d \times d}$ be an orthogonal matrix such that ...
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2answers
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If $\sum_{i=1}^n a_n=0$ then you can find a “good” ordering of $a_i$.

I'm trying to prove (or disprove, but I think it's true and I'll be surprised if someone would manage to disprove it) a small theorem. Given an array of real numbers $A=[a_1,a_2,...,a_n]$ such that ...
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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 ...
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2answers
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Interpolate/Increment Vector Rotation

For my 2D physics engine, I'm using the unit vectors of the direction an object is facing to represent its orientation; essentially, [Cos(theta),Sin(theta)] where theta is the object's rotation in ...
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1answer
26 views

Calculating plane rotation angles

Let's presume I have an arbitrary plane, for sake of simplification, centered at (0,0,0), described by coordinates of 4 vertices (and normal if needed). Is there any way to describe this plane as ...
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2answers
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How can I get if an object is facing another in degrees if one has 0 and the other one has 360?

I'm developing a model for the collision for two ships. First, I have the angle that the first ship would be pointing if it was to directly point at the second ship. I'll call the angle between the ...
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1answer
21 views

How can I divide degrees greater than 360 to be within the 360 degrees?

I've got a rotation of something in degrees, however this rotation can be greater than 360 or less than 0. How can I multiply/divide this to be within 360? For example, 1800 to be turned into 360, ...
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1answer
34 views

How do rotations work in Minkowski space?

In $4$-dimensional Euclidean space, one must take account of there being either one unique (simple), or two unique (double), or two but not unique (isoclinic) planes associated with any one rotation. ...
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3answers
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Three dimensional rotation of equations.

I have a set of equations that describe a wire in (100) direction. I want to rotate the wire such that it's in the direction (111). My initial plan (which failed) was to use Euler coordinates and ...
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1answer
27 views

3 rotation values to work out rotation in degrees

I am currently working with the Oculus headset and dealing with the Z axis. With the software I have, the values I can retrieve are limited and I was hoping someone could help me find a solution to ...
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1answer
32 views

Unit Vectors in Rotation Matrices

If I have a rotation matrix $R$, say: $$R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}$$ when I multiply it by a vector, $$V' = R.V$$ does ...
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Angle of rotation from complex eigenvalues of Rotation matrix

The complex eigenvalues of a Rotation matrix are $e^{-i\theta}$ and $e^{i\theta}$. Corresponding to these we get complex eigenvectors. We know that the eigenvector corresponding to the eigenvalue 1 ...
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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 ...
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1answer
40 views

Prove: $G = HN$, and $H \cap N =$ {$e$}, (Isometries in $\mathbb{R^2}$)

I'm trying to solve this problem: Let $G = E(\mathbb{R^2})$ be the group of all isometries of $\mathbb{R^2}$, so $G$ consists of translations, rotations about the origin and reflections in a line ...
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Find angular velocity of a varying vector

I have a vector $\vec V$ and its time derivative $\dot{\vec V}$. Knowing only this, I need to find the angular velocity of the vector $\vec V$, which we may assume as a vector turning about some ...
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1answer
43 views

Two line segments are intersecting… how do I rotate either line about the intersection point?

I have two line segments which are currently intersecting, how do I rotate either intersecting line about the intersection point? How do I calculate rotation of these lines? Rotation about a point ...
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1answer
26 views

Aligning coordinates on 2 circles

In my application I'm trying to display multiple edges between vertices in this way: Based on some calculations regarding circle segments, I have identified 4 coordinates on each circle that are ...
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0answers
35 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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2answers
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Commutativity of Spatial Rotations

I know that in general spatial rotations (rotations in $\Bbb R^3$) are not commutative. But what if we restricted our possible rotations to only those around orthogonal axes? For instance, what if ...
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1answer
25 views

How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
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1answer
21 views

2d projection of a 3d image

I am having a problem where I have a $2$D object which can move in $3$-dimensional space about a fixed point (the origin). I want to rotate this object using Euler angles and axes of rotation. If ...
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1answer
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How to get coordinates of a point after an image is rotated? (with images)

I have a problem that involves a rotating image and finding a previously known point. Firstly, there is a sequence with the rotation. We start with an empty image. A line is drawn vertically, ...
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1answer
127 views

Rotate object around a fixed coordinate axis

I am trying to let the user of my app rotate a 3D object drawn in the center of the screen by dragging their finger on screen. A horizontal movement on screen means rotation around a fixed Y axis, and ...
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0answers
28 views

net of oblique cone,why it has a shape like this?

today i was building a right cone for my geometry homework.after building the cone, i started to think what shape the net of an oblique cone (cones with circular base which the axis does not pass ...
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Problem: Minimum of set of maximum correlations of trig. poly. coef vector with space of all trig. poly. coef vectors

I have a problem that I can't seem to get started with. Let $\mathbf{c} = \{c_k\in \mathbb{C}\}_{k\le|N|}, c_{-k} = -\bar{c}_{k}, \|\mathbf{c}\|_2 = 1$ be the normalised vector of coefficients of a ...