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

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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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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
49 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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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
52 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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0answers
39 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
23 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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2answers
55 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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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
20 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
17 views

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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29 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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1answer
25 views

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
47 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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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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0answers
28 views

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
22 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
41 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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0answers
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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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 ...
2
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1answer
21 views

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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22 views

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
38 views

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}\\...
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0answers
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Rotationally invariant function in three-sphere

I know that every rotationally invariant function $f(x,y,z)$ in two-sphere $S^2$ must satisfy \begin{align*} f(x,y,z)=f(h(\theta)(x,y,z)^T) \end{align*} for all $\theta$ and $x,y,z$, where \begin{...
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1answer
28 views

Determine a point, given another point and roll/pitch/yaw

Let's assume we have a plane (or a ship, does not matter). We have a GPS-receiver installed on our plane, e.g on a wing. So, how do we find position of a point (with known shifts from our receiver), ...
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2answers
34 views

How does the Pearson correlation coefficient change under rotations

I was reading on wikipedia about the pearson correlation coefficient. Assuming the data has zero mean it can be written as $$ \rho = \frac{ \sum x_i y_i } {\sqrt{\sum x_i^2 \sum y_i^2}} $$ The ...
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Optimal ordering in Jacobi SVD algorithm

In Jacobi SVD algorithm as given here every pair of columns of the matrix is orthogonalized until convergence. I want to know that how does the order of selection of the pair of columns affect the ...
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3answers
45 views

is the rotation matrix is unique for one rotation

I have a test for rotation , and found two rotation behave the same at one point ...
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3answers
55 views

Calculating the distance in degrees between two points on a circle

In a video game I am I'm working on I'm trying to rotate an object around a secondary object. The secondary object will always be in the exact center and the rotating object will always rotate in a ...
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1answer
318 views

Can any higher-dimensional Spheres be rotated everywhere equally?

You can rotate a circle so that every point on it (just the perimeter, not the interior) moves "equally". That is, every point moves with the same speed and even has the same "acceleration" (first-...
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1answer
32 views

Notation: rotation matrix with a condition

I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused: Is the top part ...
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28 views

Projection from high dimension to lower, for visualization

I want to project high dimensional data points onto 2D screen coordinates, for visualization purposes. I want to be able to control the angles of projection manually (eg, with the mouse). I have ...
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1answer
46 views

Can quaternions be used to represent rotation rate?

A quaternion is a useful tool for representing a rotation, or change in attitude. If a quaternion $q$ represents a rotation, and $v$ a vector, then $v'=qvq^*$ rotates the vector, where the multiply ...
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2answers
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How to “rotate” a function? Or, how to write a function which has a known, rotational symmetry with respect to another function?

EDIT 2: I've posted my "real" question here: http://mathematica.stackexchange.com/questions/115766/finding-closed-form-eigenvalues-of-a-particular-matrix I have posed my question formally in LaTeX ...
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1answer
19 views

Perform a rotation in 3D world

I got a character at some point $A$ facing to point $O$ that is equal to $(0,0,0)$, then I move it to point $B$ and I want to rotate him to face point $O$. Since this is 3D world I think that I need ...
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2answers
38 views

Angular Difference between two rotation matrices on XZ plane

As the title says, I have two rotation matrices, $ R_1 $ and $ R_2 $. Both are rotation matrices that transform from the origin coordinate system $O$ to positions $1$ and $2$ (ignoring any translation)...
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I have this seemingly simple volume of a solid of revolution, but the limits and function are unknown.

How can I possibly find the numerical area of the region without knowing the function itself or the limits?
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2answers
804 views

Modelling the “Moving Sofa”

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
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1answer
18 views

How to calculate rotation rates of a rotating body relative to another rotating body?

I have two 3D bodies A and B, each of them is rotating around its own Z-axes with an angular velocity (e.i. yaw rate) of $\dot{\alpha}_A$ and $\dot{\alpha}_B$, respectively, relative to an absolute ...
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1answer
20 views

How to calculate rotation quaternion between two orientation quaternions?

I have some device (3D pointer) connected to my computer which returns it's position (in cartesian XYZ system) and orientation (in quaternions). I receive this values about 30 times/sec. Now I need ...
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1answer
45 views

Symmetric Matrix in SO(3) : Exponential Formula

Let $R\in $ SO(3), that is $R$ is real $3\times 3$ orthogonal matrix with determinant $+1$. I am trying prove that if $R= R^\top$, and $R\in $ SO(3) then $R \in \{exp(k\pi \hat{a}) | k\in \mathbb{Z}, ...
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1answer
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Rotation of a plane

Parc c of Exercise 4.3.12 in Shifrin and Adams' Linear Algebra: a Geometric Approach says Let $V \subset \mathbb{R}^3$ be the subspace defined by $$V = \{(x_1, x_2, x_3): x_1 - x_2 + x_3 =0\}.$...
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Rotation of axes

Is it possible to rotate the axes along a point that is NOT lying on the axes? For example, consider the point C(u,v) where $u,v \neq 0$. Can I rotate the axes about this point? In my mind, this ...
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1answer
50 views

Square's angle of rotation using 4 corners

I need to calculate angle of rotation of an image from X and Y axis, what I'll have is 4 co-ordinates of the corners of the image (basically a perfect square image is rotated in all possible angle), ...
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Why don't more celestial bodies exhibit higher-order rotations?

It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have ...
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Vector on a sphere

I have for some time tried to understand the math behind explained in this post, but seem to not grasp. I think the way i visualize it might be incorrect, which make harder for me to grasp what is ...
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1answer
38 views

Quaternion to Euler angles conversion

I have written the following MATLAB code for transforming Quaternion to Euler angles based on the mathematical formula from wikipedia: ...