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

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Finding normal of arbitrarily oriented ellipsoid

For an axis-aligned ellipsoid, the equation is $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1. $$ With $a=b$ this will give a spheroid with the $z$ axis as its symmetry axis. A spheroid ...
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16 views

Individual components of a quaternion?

My 3D system consists of a global coordinate frame and a quaternion in this frame represents a roll (about x), pitch (about y) and yaw (about z) in the form of four values (qx, qy, qz, qw). How can I ...
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18 views

Subtracting Direction cosine matrices

Say I have a rotation matrix $C(q) \in R^{3\times3}$ which is a function of a Hamilton quaternion $q=[q_w, q_x, q_y, q_z]^T=[q_w, q_v]^T$ where $q_v$ is the vector part of the quaternion. The ...
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2answers
34 views

How to rotate a line based dimensions of a piece of paper

I have a line where I know the start and end point on a piece of paper with the dimensions of 8 1/2 inches x 11 inches. the start point is 5.6 inches from the right of the paper and 4 inches down ...
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2answers
61 views

Decomposition to rotation around arbitrary axis

In 3d, I have a $4\times4$ matrix $M$, which has only a rotation part and a translation part. In other words, I can compute $X'=RX+T$ ( with $R$ a $3\times3$ rotation matrix, $T$ a vector for the ...
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1answer
40 views

Can Someone help me with my trigonometry rotation, formula? [closed]

I've been working on some code for a game to make a hit box, this question is just about the math though. Basically I'm trying to rotate an X, Y point(i guess according to the game it's Z,X Not sure ...
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2answers
64 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
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2answers
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Quaternion Rotation

I am modelling rotations of a rectangular box (3 dimensions) in Matlab using Quaternion theory. Using the theory found on https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation I have ...
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1answer
29 views

matrix multiplied by rotation matrix on right side and transpose(rotation) on left side

Would a matrix remain un-rotated if it is multiplied by an orthonormal rotation matrix on right side and transpose of same rotation matrix on the left side?
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2answers
77 views

The relation between axes of 3D rotations

Let's suppose we have two rotations about two different axes represented by vectors $v_1$ and $v_2$: $R_1(v_1, \theta_1)$, $R_2(v_2,\theta_2)$. It's relatively easy to prove that composition of ...
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1answer
20 views

Inverse Parameters of a Pan-Tilt Rotation Possible?

I have a 2-parameter (tilt,pan) rotation computed as tilt followed by pan, i.e. two rotation matrices multiplied together: $$R(t,p)=\begin{pmatrix} c_p & s_p s_t & s_p c_t \\ 0 & c_t &...
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0answers
37 views

Sun path formula

How can I calculate the unit vector to the sun for specific time relative to observer position, given with lat:lon coordinates, considering Y+ axis directed to North, X+ axis directed to East and Z+ ...
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0answers
14 views

Rotation of basis vectors to new orthogonal axes

I have a set of orthogonal axes described as three vectors, which relate to the principal axes of an ellipsoids. My question is how do I calculate a rotation matrix, or Euler angles, which rotates an ...
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1answer
41 views

2nd order derivative of Lie group SO(3)

In P.4 of this technical report there is a equation: \begin{align} \left.\frac{\partial^{2}}{\partial \omega_{x}\partial\omega_{y}}(\mathbf{R}_{0}\exp\{J(\omega)\}) \right|_{\omega=0} & = \...
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1answer
25 views

Rotate and transforming in one step

This comes from a programming problem from CodeAbbey, however, I cannot figure out the mathematical formula to solve the problem. Within the problem here we have a graph, then the same graph ...
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28 views

Question about a rotation around a line in $R^3$.

Say I have a linear application $\phi: R^3 \rightarrow R^3$ that is a rotation around the line $x_1 - x_2 = 0 , x_3 = 0$ How can I deduce that the first two columns of the matrix associated with the ...
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2answers
23 views

Parametrized equation of hyperplane orthogonal to main diagonal

For the hyperplane passing through the origin and orthogonal to the ones vector,$(\underset{n\ \mbox{times}}{\underbrace{1,\dots,1})}$ in $\mathbb{R}^{n}$, what are the $n-1$ remaining orthonormal ...
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2answers
121 views

Symmetrical and skew-symmetrical part of rotation matrix

Every matrix can be decomposed to symmetrical and skew-symmetrical part with the formula: $ A=\dfrac{1}{2}(A+A^T)+\dfrac{1}{2}(A-A^T)$. However if it is known only symmetrical part (we assume here ...
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2answers
45 views

translation and rotation of a parabola

I am trying to translate a parabola to the origin, rotate by T radians and then translate back to the original position. I can calculate the new X and Y vectors using matrix operations and the regress ...
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0answers
28 views

How to rotate an $n$-dimensional normal distribution, to maximize the likelihood of a sample

Suppose we have a normal distribution with a diagonal covariance matrix S and mean $0$, i.e. $N(0,S)$. I want to find a Rotation matrix $R$, to rotate this distribution to maximize the likelihood of a ...
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16 views

Transformation of rotation between fixed frame and rotating frame

Sorry for boring you my friends. I am haunted by a problem of rotation. Suppose that we have two frames $F_1\left( \bar{X},\bar{Y},\bar{Z}\right)$ and $F_2\left( {X},{Y},{Z}\right)$. $F_1$ is the ...
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Proof of the rotation matrix is an extreme point of $\text{conv } SO(n)$

Define the set of rotation matrices: \begin{equation} \begin{aligned} SO(n) := \{X\in \textbf{R}^{n\times n}: X^TX=I, \text{det}(X)=1\} \end{aligned} \end{equation} I want to prove that if $X\in SO(...
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27 views

Find rotation matrix to match points in parallel projection

I am given two sets of 3D points (actually 2D, see below) with corresponding pairs. I am seeking two 3D rotation matrices, such that (only) the X and Y components of the rotated points match best (...
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4answers
67 views

The formula for 3D rotation of the perspective of an image in 2D space

Do you know what the formula is for the 3D rotation of the perspective of a 3D object in a 2D space. For example, imagine that we got a picture of a 3D object. So, we have the projected picture of ...
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1answer
71 views

The maximal rotation matrix

Let's consider two numbers calculated for a rotation matrix which are: $s_e=$ the sum of all entries of a matrix $s_a=$ the sum of absolute values of all entries for a given matrix. It ...
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0answers
24 views

rotating a point using a previously rotated one

I want to rotate a shape in an n dimensional space (n>3) around (about) the origin. knowing the outcome of rotation on a point like A, which is A', how can I find the rotation outcome on a point like ...
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2answers
55 views

If $f(x)=x^3-3x^2-x+2$ ,Find an expression for the function $y$, which is obtained by rotating the graph of $f (x)$ through $180°$.

Question: $f(x)=x^3-3x^2-x+2$ where $x\leq 1$ Find an expression for the function $y = g(x)$, where $x\geq1$ , which is obtained by rotating the graph of $y = f (x)$ through $180°$ about ...
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2answers
515 views

Additive rotation matrices

Let's assume that we want to find a rotation matrix which added to a given rotation matrix gives also a rotation matrix. I would name such matrix a rotation additive matrix for a given rotation ...
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44 views

Why is this Isometry an rotation?

i need a little help. Did someone have an idea how to prove this? Thanks in advance. Be $\Phi$ an direct isometry of the euclidean Space $\mathbb{R}^3$ with $\Phi (\left(\begin{eqnarray} 2\\0 \\1 \...
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1answer
29 views

Improper rotation matrix in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
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64 views

How to calculate matrix rotation

Given the following rotation matrix $$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{matrix} \right]$$ ...
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32 views

can somone exaplin how the equations for quanterion roation actually work, or at least show me some?

I don't know enough about qunaterion rotation to as the question easily. Sufficient to say I have a system were three rectangles on a 2 axis plan (I think those words are right.....) that are ...
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1answer
22 views

orientation from acceleration?

I have an object that has a sensor attached on it. This sensor calculates the acceleration in all axis and angular acceleration in all axis(Keep the gravity force in mind). How can I get the Pitch, ...
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1answer
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Converting a geodesic into a set of Euler angles

Two non antipodal points on a sphere have a geodetic which is a segment of a great circle on that sphere. I'm trying to calculate the Euler angles that would rotate the "equator" great circle of the ...
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30 views

Multiplying quaternions vs multiplying rotation matrices

It's a trivial question, but one I'm not 100% clear about. Given two matrices $$P_{\{1,2\}} = \left[ \begin{array}{cc}R & t \\ \textbf{0} & 1 \end{array}\right]$$ where $R$ is a 3x3 ...
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3answers
180 views

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \...
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1answer
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What is the angular velocity in an inertial frame given the angular velocity in a body fixed frame?

At a given time t, the rotation matrix R has the value: $$R= \begin{pmatrix} 0.675 & −0.1724 &0.7174\\0.2474 & 0.9689& 0 &\\−0.6951& 0.1775&0.6967. \end{pmatrix}$$ The ...
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what is the ZYZ euler angle representation,(ψ, θ, ϕ), for a given matrix in MatLab?

I want to know using matlab what is the ZYZ euler angle representation,(ψ, θ, ϕ), for the following rotation matrix? \begin{pmatrix}0.6927&-0.7146&0.0978\\ \:0.7165&0.6973&0.0198\\ \:-...
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1answer
49 views

How to verify that a matrix is a rotation matrix in Matlab?

Using Matlab, I want to know if $$A=\begin{pmatrix} \cos(x) & \sin(x)\\ -\sin(x) & \cos(x) \end{pmatrix}$$ is a rotation matrix. Hence, $$\begin{pmatrix} \cos(x) & \sin(x)\\ -\sin(x) &...
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29 views

Calculating a rotation matrix: error and properties

Given an axis $u=\begin{bmatrix} \sqrt 3/3, \sqrt 3 /3, \sqrt 3 /3 \end{bmatrix}$ and an angle $\phi =\frac{2\pi}{3}$ I want to calculate the related rotation matrix: Well given Rodriguez’s formula: ...
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1answer
17 views

Verifying that the Rodrigues formula gives the same result as $Rot(x,\phi)$?

How to verify that the Rodrigues formula with $x$ as an axis of rotation and $\phi$ the angle of rotation with $u$ a unit vector along $x$ and $Rot(x,\phi)$ gives the same result? I only know that ...
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Euler Angle + Distance to XYZ coordinate

Background I'm familiar with Euler Angles and 3d space systems but I'm having trouble with Rotation Matrices. Scenario I've converted my Euler Angle to degrees for ...
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1answer
52 views

Mathematical expression of a rotation

I don't understand how my teacher defined an expression for the rotation adding up the two red vectors made up from the strong blue ones after rotation I especially don't understand how does the ...
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1answer
25 views

Transformation of fourth rank tensor and its matrix form

I would like to calculate transformation of fourth rank tensor, $$ C_{ijkl}=\Sigma_{m=1}^{3}\Sigma_{n=1}^{3}\Sigma_{p=1}^{3}\Sigma_{q=1}^{3}a_{im}a_{jn}a_{kp}a_{lq}C_{mnpq} $$ where $a_{xy}$ is ...
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2answers
43 views

Quaternion interpolation in 3D

I'm a chemist lost in the captivating world of mathematics thus if you could keep your answers simple it would be awesome! Here is my problem: I have two mobiles (A,B) in 3D. Ideally, I would like to ...
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33 views

What is the shape of an ellipse (or parabola) that has rotated around the x-axis?

What is the shape of an ellipse (or parabola) that has rotated around the x-axis? Is there a specific name?
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42 views

Alignment of one 3D Coordinate system to another 3D Coordinate system

I'm working on a project depicted by this picture(taken from internet) where there are different coordinate system involved which corresponds to camera coordinate system and local 3D coordinate system ...
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1answer
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The meaning of spacecraft attitude represented in quaternion

I am reading the following paper about the attitude control of aircraft: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1271671 The quaternion represents the relative orientation of two ...
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Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
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3answers
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Confusion in rotation matrix - rotation about $y$ axis

The rotation matrix about y axis should look like $$\left[ \begin{array}{ccc} \cos\frac{\pi}{2} & 0 &\sin\frac{\pi}{2}\\ 0 & 1 & 0\\ -\sin\frac{\pi}{2} & 0 &\cos\frac{\pi}{2}\\...