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

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1answer
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Rotation Matrix and programming [on hold]

I am actually programming in Android. An android tablet as a lot of sensors including one that gives the rotation vector of the tablet. (See ...
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2answers
40 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
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0answers
13 views

how to convert eigenvectors & eigenvalue to rotation matrix?

I would like to know how to convert an eigenvector and an eigenvalue(if needed) to a rotation matrix. I am in charge with writing software to calculate the attitude of a satellite in space. K is a 4 ...
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0answers
34 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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0answers
10 views

How can I define the order of rotation from one rotation matrix to another?

I am trying to rotate a koordinatesystem in the defined order of z-x-y. I have the matrix $rot_x$ describing the rotation around the x-Axis, the matrix $rot_y$ describing the rotation around the ...
0
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1answer
22 views

how to rotate scaled-vector (orientation) by scaled-vector (rotation)

Recently I got the physics-engine portion of my 3D simulation / game engine working correctly. The most convenient way to store and compute position and orientation are in 3-element vectors (though my ...
3
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1answer
53 views

free groups of rotations

The question of which pairs of rotations of the sphere are independent goes back to Hausdorff, who produced such a pair a century ago. "Independent" means "are free generators of a free group". The ...
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0answers
12 views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
2
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0answers
16 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 ...
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0answers
32 views

Transformation matrix for two rotations

If I am required to compute the full transformation matrix compromising of the following sequence of operations: rotation by $30$ degrees about $x$-axis translation by $1$, $-1$, $4$ in $x$, $y$ and ...
1
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1answer
35 views

3D rotation of an object with respect to another object's rotation

I am writing a python code to translate and rotate an object with respect to another object. Please take a look at the picture bellow: The smiley face and the arrow have initial poses (position ...
3
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3answers
56 views

How many degrees of freedom would a rotation matrix in R5 have?

I understand that a rotation matrix in R3 has three degrees of freedom because there is three linearly independent planes that the rotation can take place in. How does this translate to R5?
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0answers
39 views

How do I get the rotation between two rotationmatrices?

I am stuck on a little rotation problem. The problem: I have 2 rotation matrices $A$ and $B$. $A$ and $B$ are relative to the coordinate system O. $A$ and $B$ are Quaternion rotation matrices. I am ...
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1answer
13 views

Counterclockwise rotation matrix

If I take the basis $(\vec{e_x},\vec{e_y})$ and make a rotation counterclockwise of angle $\theta$, I end up with two new vectors $(\vec{u},\vec{v})$ such that : $\vec{u} = \cos\theta \vec{e_x} + ...
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0answers
27 views

rotation matrix - satellite control system

I am in charge with developping a satellie control system. There is something that I either understood incorrectly or either is wrong in the explanation. I am checking if I am able to get the values ...
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0answers
11 views

two rotations of line segment in different order

there are given are two rotations: $R_1$ and $R_2$, and a line segment $AB$. the image of $R_1R_2(AB)$ is a line segment that is parallel to the line segment $R_2R_1(AB)$. my kindly request is to the ...
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0answers
15 views

Education tool for learning 3D angles

I hope it is not an off-topic. I have started working on 3D frame transformation and transforming a vector such as acceleration or angular velocity from one coordination to earth coordination. My ...
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0answers
10 views

The z-axes of two orthoganal cartesian coordinates frames are aligned, then rotated about their x-axes by an angle. How do I calculate that angle?

I am trying to reverse a series of rotations applied to some Cartesian coordinate systems. Two coordinates systems, C1 and C2, are originally oriented with their z-axes aligned but not their x-axes ...
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0answers
30 views

Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec ...
0
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1answer
26 views

Rotation in high dimension in the direction of given vectors

Given two vectors $A$ and $B$ (with high dimension), and an angle $\alpha$. How can one find the vector $C$ which is $A$ rotated over $\alpha$ in the direction of $B$? If it changes anything: the ...
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2answers
20 views

Characterize a rotation matrix

Given a matrix $A\in M_{2 \times 2}(\mathbb R)$ or $M_{3\times 3}(\mathbb R)$ how to determine if it is a rotation matrix? Is there any theorem that characterize a rotation matrix just by looking at ...
2
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1answer
36 views

High Dimensional Rotation Matrices As Product of In-Plane Rotations

Lately I've been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? ...
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3answers
53 views

Rotate an area around a diagonal line.

I know how to find the volume of the figure formed when you rotate a $2$-dimensional area around a horizontal or vertical line, but what if it were a diagonal line instead? For example: Rotate the ...
2
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2answers
30 views

how to combine angle rotations along different axes into one rotation along a single vector [duplicate]

So, lets say I have some rotation a about the x-axis(vector:$(1, 0 ,0)$) and some other rotation about y-axis(vector $(0, 1, 0)$) and a rotation about the z-axis(vector: $(0,0,1)$). How would I ...
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2answers
31 views

Caclulate X,Y coordinates of point after rotation around another point of given degrees

There are Two Points A and B. The linear distance between the points is R. I have the ...
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1answer
44 views

Rotation Matrix with (Cos(theta) = 0,Sin(theta) = 1) as Identity

In my program, all rotations are handled with unit-vector orientations: $$[x,y] = [\cos{\theta}, \sin{\theta}]$$ In the game engine I'm using for visualization (Unity3D), the $Y$ axis is forwards - ...
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0answers
34 views

Rotation that is swapping quaternion $w$ and $x$, $y$ and $z$

I have a quaternion, representing a rotation, equals to: $(w, x, y, z)$ If I convert that quaternion to euler angles, add 180 degrees to $x$ and $z$, and convert it into a quaternion again, I get : ...
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0answers
19 views

Using Levi-civita symbol to determine axis and angle of rotation matrix

One of the questions on the course involves finding the angle and axis of this rotation matrix; $$R = \gamma\ \begin{pmatrix} 0 & -2 & 1\\ 2 & 0 & 0\\ -1 & 0 & 0 ...
3
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1answer
37 views

Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$

Question: Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$ where $A = (0, 0), B = (2, 0), C = (1, −2), D = (1, 0)$. Before, I jump in ...
1
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1answer
74 views

Relative distance to rotated object

An object (with center $O_{2}$) has been rotated by an angle $\alpha$. There are two images of the object taken by a camera (centered at $O_{1}$), and two points $x_{1}$ and $x_{2}$ that are actually ...
2
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1answer
23 views

Reflection combined with a glide reflection

Suppose that $A, B, C,$ and $D$ are the vertices $(0, 0), (2, 0), (2, 2),$ and $(0, 2)$ of a square. The transformation $ρ_{AC} ◦ γ_{DA}$ can be decomposed as the combination of reflections across ...
1
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1answer
31 views

Expression of rotation matrix from two vectors

What is the matrix expression of the rotation matrix in 3D which turns a vector $\vec{a}$ into a vector $\vec{b}$, with both vectors given by their coordinates? ($\vec{a} = (a_x, a_y, a_z)$ and ...
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2answers
50 views

Differentiation of a Vector with respect to a vector

I am studying a paper and I am going crazy about one differentiation which it is written on it but not explained. I think I am missing something and probably something easy. I would love if someone ...
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0answers
12 views

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 ...
1
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1answer
33 views

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
20 views

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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0answers
19 views

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 ...
2
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3answers
63 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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0answers
10 views

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 ...
0
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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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0answers
12 views

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
46 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
47 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
11 views

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
19 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
15 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 ...
3
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1answer
20 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 $R \circ P$, ...
0
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1answer
13 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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0answers
23 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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0answers
46 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 ...