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

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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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0answers
10 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
18 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
33 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 ...
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2answers
28 views

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
687 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
17 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
16 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
42 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
21 views

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 ...
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25 views

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
38 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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0answers
41 views

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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0answers
23 views

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
30 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: ...
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1answer
7 views

Getting Tait-Bryan Angles from Quaternion for a Non-Standard, Left-Handed Coordinate System

I am trying to write a autopilot script for Kerbal Space Program, which requires me to do some conversions between Tait-Bryan angles and quaternions. Unfortunately, KSP uses a left-handed coordinate ...
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2answers
26 views

Difference between quaternions depends on initial rotation

The difference $\Delta q$ between two quaternions $q1$ and $q2$ can be calculated as $\Delta q = q1\cdot q2^{-1}$, where $^{-1}$ is the quaternion conjugate. When numerically evaluating the ...
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1answer
33 views

Rotation about z axis using quaternions

I am working with quaternions and rotation, but I am missing something about how a rotation expressed as a quaternion works. I also discovered that there are different convention for quaternions (JPL, ...
0
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2answers
57 views

How do I rotate a vector 90 degrees in a random direction?

I'm building a tree generator and I'm at the point where I want to have sub branches branch off at right angles off the current branch, in random directions. I have a 3D vector defining the direction ...
0
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2answers
41 views

Quaternion angle - Opengl rendering

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). I am trying to calculate the angle of rotation around all the three axes and Render a 3D cube using opengl to immitate the ...
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2answers
24 views

Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
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2answers
38 views

$2x^{1/4}$ rotated around $y = 2x$

This is the question: find the volume created by rotating $2 x^{1/4}$ around $y=2x$. I was able to define the distance between the two lines as $y/2 - (y/2 )^4$. However, I can't find the radius that ...
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0answers
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Show that $δ_{KL}$ is a Cartesian tensor

By using the definition of the Kronecker delta $δ_{KL}$, show that $δ_{KL}$ is a Cartesian tensor, that is $δ'_{MN} = L_{MK}L_{NL}δ_{KL}$ under the rotation $X_K = L_{MK}X'_M$. Solution: Using the ...
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0answers
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In a transformation matrix, why is $Y$-axis ($-\sin$) in left most column as opposed to right like X and Z [duplicate]

$4 \times 4$ Transform Matrix with axis columns $XYZ$ left to right $X$-axis rotation: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & \cos & -\sin & 0 \\ 0 & \sin & \cos ...
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45 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
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2answers
48 views

Prove that 3d rotation is linear

In a 2d space, a transformation is linear if $f(v+w) = f(v) + f(w)$ and $f(kv) = k*f(v)$, and rotation preserves addition so it is linear. In a 3d space, similar rules apply: $(x, y, z) + (l, j, k) = ...
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1answer
11 views

Estimate angular velocity and angular acceleration from a point cloud sequence

Lets say I have the a set of points $P = \{p_1, p_2, ...\}, p_i \in R^3$ that change position with time. These points are part of a rigid body and I record these positions in order to estimate its ...
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0answers
80 views

Rotating one 3d-vector to another only by using rotations about the coordinate axes.

If I have a vector v=(x,y,z) and would like to transform another vector u by using only rotations about the coordinate axes to be in the direction of v, how can I find required angles and the order of ...
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1answer
34 views

How to decompose a unit quaternion into 3 Tait-Bryan quaternions instead of 3 real numbers?

I'm familiar with the formulas for decomposing a unit quaternion $Q$ into chained Tait-Bryan angles $\phi\theta\psi$ (Wikipedia has the formulas for the $zyx$ chain here), but I'm looking to instead ...
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1answer
16 views

Rotations about the origin

Let R(θ) denote a rotation matrix which rotates a point $x$ in $S^2$ anticlockwise about the origin through a given angle θ. (Where $S$ is the set of real numbers) How do you illustrate that this ...
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2answers
70 views

Calculate Rotation Matrix to align k n dimensional vectors

I have a $k$ number of $n$-dimensional vectors written with respect to two rotated frames: $X= \{\vec{x}_1,\vec{x}_2,...,\vec{x}_k\}$ and the same rotated vectors: $X'= ...
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1answer
63 views

Why does rotation by a quaternion require multiplying two times?

Given a vector $p$, to rotate it by a quaternion $q$, we use the formula: $$p' = q p \hat{q}$$ where $\hat{q}$ is the conjugate of $q$. But if we use rotational matrices, then it's just $$p' = ...
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27 views

Convert a 3d vector into a rotation matrix?

Is it possible to compute a Rotation matrix given a 3d vector given in the Euclidean space? and if not what would it need? An illustration of my situation. Illustration of my problem I have a ...
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25 views

How do i compute how much i can rotate my tool?

I am at moment trying to implement an Ball tracker for a robot arm with a stereo camera monted on it as its tool. Illustration: http://m.imgur.com/5oojXdh The camera provide me with an dx, dy, dz ...
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17 views

How do i convert an x,y,z to an Q configuration?

I am trying to implement a tracking application for a robot arm, which purpose is relocate itself based on the position of an object seen from the tool point. illustration: http://imgur.com/5oojXdh ...
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1answer
19 views

Rotations around an axis

I am given a linear transformation $T:\mathbb{R^3} \rightarrow \mathbb{R^3}$ The transformation is linear and is defined by taking a vector in $\mathbb{R^3}$ and rotating it around the axis ...
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1answer
19 views

Applying Rotation Matrix to Rotate Point (2,4,4) wrt y onto X-Y Plane

My linear algebra is a little rusty. Given an arbitrary point P(x,y,z), how do I determine the theta needed to apply a rotation matrix that will rotate the point onto the X-Y plane, with respect to Y? ...
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24 views

Graphing calculator leaving gaps in a drawn graph of a rotated parabola

I am graphing the equation of two rotated parabola on the graphing calculator and, after finding the y= form for each using the quadratic formula and entering them into a program to graph them they ...
1
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1answer
36 views

Calculating a quaternion that represents a given rotation

This is the first time I'm attempting to do a quaternion and I am not quite getting the concept. This is part of a 3 calculation homework question The initial question is Given a 3-D point at ...
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2answers
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Proof there isn't a vector u such Su=u where S is the rotation transformation in R2 [closed]

We have the rotation matrix \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{pmatrix} Proof: there isn't exist a vector $u \in\ {\mathbb{R}^2}$ ($u\neq0$) such ...
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1answer
29 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert ...
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9 views

How do we rotate a vector in a 3D cartesian plane with the plane of rotation specified?

I know the rotation method of a complex number that : "If we multiply a complex number $x + \iota y$ by $e^{\iota \theta}$ where $\theta$ is the angle of rotation in anticlockwise sense then we get a ...
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If $A$ is a rotation matrix by $\theta$, then what does $A^T$ do?

Little help here? If multiplication by $A$ rotates a vector $x$ in the $x$-$y$ plane through an angle $\theta$, then what is the effect of multiplying $x$ by $A^T$. Explain your reasoning. Any help ...
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How to calculate coordinates $(x,y)$ of rotated polygon?

I have coordinates $x,y$ of a point (red point on the image). If I rotate the image with a specific angle (for example 30 degrees) how can I get the coordinates $x,y$ in the new polygon (which is ...
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1answer
25 views

3D calculate new location of point after rotation around origin

I've tried to boil down my problem as much as possible. I've got two questions, but really I'd be satisfied enough just knowing how to accomplish the first one. I'm looking to do this programatically, ...
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0answers
16 views

Rotate the plane through 90 degree?

I have a problem from the book "Linear Algebra and Its Applications" by G. Strang. What $3$ by $3$ matrix represents the transformation that rotate the $x$-$y$ plane through $90^{\circ}$, ...
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2answers
16 views

Robustly map rotation matrix to axis-angle

The Wikipedia article for rotation matrix gives the following formula for converting from rotation matrix, $Q$, to axis-angle, $u$ and $\theta$: $$ \begin{align} x &= Q_{zy} - Q_{yz} \\ y &= ...
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1answer
31 views

Why 90 degree rotation form a special case in Euler angle representation of rotation

I was reading a document illustrating euler angles representation of rotations. A Rotation that depends on 3 angles $\alpha$ around z axis, $\beta$ around y axis, and $\gamma$ around x axis is ...
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63 views

Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...