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

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What rotation rules can be applied to stacked cubes to make a 3D spirograph?

If you arrange building blocks for example toy cubes so that every next cube is tilted over its base by 30 degrees and rotated to it's right by 12 degrees, it would wind through space in a helical ...
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Euler's rotation theorem [closed]

Say I have two coordinate sytems, A and B. I take B and rotate it about B's Y axis, and then about B's new Z axis. From the point of view of A, B's Z axis just rotated about Y. So the Euler axis ...
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1answer
16 views

Transformation of general equation of second degree with respect to a rectangular axes

Question : The equation $3x^2+2xy+3y^2-18x-22y+50=0$ is transformed to $4x^2+2y^2=1$ with respect to a rectangular axes through the point $(2,3)$ inclined to the original axes at an angle $\theta$. ...
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33 views

Rotation in 4D?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is ...
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1answer
35 views

Finding Moment of Inertia of a Rugby Ball

I am asked to compute the moment of inertia about the $z$-axis of a rugby ball in terms of its total mass. A rugby ball surface is given by the ellipsoid: $$\frac{x^2}{4} + \frac{y^2}{4} + ...
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0answers
33 views

How to rotate two 2D ellipses such that they have maximum cross corelation?

Suppose I have two matrices $A=\begin{pmatrix}3 & 1 \\ 1 & 4\end{pmatrix}$ and $B=\begin{pmatrix}5 &-2 \\ -2 & 4\end{pmatrix}$, where $A$ and $B$ represent covariance ellipses in 2D. ...
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1answer
50 views

How to prove a lemma required for the Banach Tarski Paradox?

I tried to teach myself the proof of the Banach Tarski Paradox by reading Terence Tao's paper on the subject; the link to the paper is here: ...
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19 views

How can obtain curvature from the gradient of a rotation tensor?

I have proper rotation function $R$ over $\mathbb{R}^3$ that yields a $3 \times 3$ tensor $R(x,y,z)$ for every point (x,y,z) in space. If I differentiate this tensor with respect to position (x,y,z), ...
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4answers
34 views

Calculating Rotation from centroid

I have a polygon as such: where the green polygon is the rotated polygon and the purple is the extent of the polygon. Is there a way to calculate the angle of rotation of the green polygon from the ...
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0answers
50 views

Quaternion rotation intuition

Say the quaternions real and imaginary part are written as $(q_1, \vec q)$. One useful multiplication property is $qr=(q_1r_1 - \langle\vec q, \vec r\rangle, q_1\vec r + r_1\vec q + \vec q \times \vec ...
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1answer
26 views

Finding rotated orthogonal vectors without knowing lengths

I have two abstract orthogonal vectors $\mid a\rangle$ and $\mid b\rangle$: $\langle a\mid b\rangle=0$, but I don't know the lengths $\mid a\mid=\sqrt{\langle a\mid a\rangle}$ and $\mid ...
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1answer
21 views

Calculate the X Y and Z rotations from one vector to another

If I have a $3$ vectors representing the $X$, $Y$ and $Z$ Axis of an object how would I calculate the rotations needed to get to that point from its original position of \begin{align} \text{$X$-axis ...
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1answer
39 views

A proof that an orthogonal matrix with a determinant 1 is a rotation matrix

Reading proof(starting on page 5) for item 1 of "Rotation Matrix Theorem" in this doc i'm stuck at understanding its last step. Matrix A being an orthogonal Matrix, at this step the conclusion that A ...
2
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2answers
39 views

Concise description of why rotation quaternions use half the angle

I'm currently writing the report on my master thesis project, where I use Android sensors to perform inertial navigation in a heavy industrial environment. In my application, I make use of quaternions ...
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0answers
7 views

How can I do a longitude/latitude tilt transformation?

I am trying to find a way to express the shortest path between two random points on a globe as a function expressed in longitude/latitude without using the geodesic equation (because it's messy and I ...
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0answers
31 views

Decomposing a matrix as the product of rotations

I'm reading an article about joint diagonalization algorithms. The article states without proof that any nonsingular $n \times n$ matrix $Q$ can be decomposed as \begin{align*} Q = \prod_{1 \leq p ...
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1answer
45 views

What does the subspace of SO(3) corresponding to zero yaw look like

Background : I'm solving an engineering problem where I have to estimate the orientation of a body in 3D space. Usually, I use quaternions to do this, but I have to consider a special case where I ...
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1answer
15 views

Convert two orthogonal unit vectors into euler vector

I have two orthogonal unit vectors that would correspond to an orientation of the Z and Y axes. I want to convert this to a rotation/Euler vector. In other words, I want to convert between two ...
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2answers
26 views

Geometrical calculation to enlarge the height of rotated rectangle

There is a polygon (rotated rectangle) that defined by 4 corner points in 2D coordinate system. Does anyone help me with the fast (minimum trigonometry operations) algorithm to change its height by ...
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0answers
29 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
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0answers
18 views

Determine mutual location of two coordinate systems, given two sets of points

My problem is: we've got tracking device and a robot. Tracking device provides set of $n$ points in cartesian coordinates(taken from marker on robot arm) and robot driver returns position of TCP(tool ...
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1answer
51 views

Quaternions and rotation

Basically, I am programming an iOS application where I use attitude of the device in quaternion format. Problem is following: Practically: I have a device that does a measurement #1 of magnetic ...
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Transformation from unknown orientation representation to DCM

I'm working with some really strange software which has some sort of custom orientation representation, and I'm trying to get it into a standardized format (direction cosine matrix). However, that's ...
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0answers
62 views

How can I use the Bullet-Physics's ray-cast normal to calculate angles for a object to lay on a surface?

[Give the normal of a surface in XYZ format, how do I calculate rotations (also in XYZ format) needed to set an object parallel to the surface?] I have a collision library that uses the bullet ...
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2answers
49 views

How to rotate a whole rectangle by an arbitrary angle around the origin using a transformation matrix?

Suppose, I have a 2D rectangle ABCD like the following: $A(0,0)$, $B(140,0)$, $C(140,100)$, $D(0,100)$. I want to rotate the whole rectangle by $\theta = 50°$. I want to rotate it around the ...
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1answer
42 views

Perpendicular vectors in $\Bbb R^3$

Hi I am struggling with this simple question. Let $\vec{v}$ be a unit vector in $\Bbb R^3$. How can I construct two periodic functions $\vec{x}(\theta)$ $\vec{y}(\theta)$ such that $\vec{v}$, ...
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23 views

Reversing a rotation around an offset center of rotation

The best way to generally phrase my question is that I have a sphere offset from its center of rotation and a vector between the sphere and a target object at a known $(\theta,\phi)$ on the sphere. ...
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0answers
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Finding points inside of a box

I have a set of points in 3D that define a large, complex object. These points are rendered in OpenGL for an Android app that I am programming. In this app, the user translates the center of the box ...
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0answers
52 views

Rotating a 3d-ellipse equation?

So I have very limited Linear Algebra knowledge, and I'm trying to program a computer graphics application in Android using OpenGL. I understand my design is not great, so if you have questions as to ...
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29 views

Get the coordinate offsets with known rotation angles (i.e. Yaw, Pitch, Roll)

I'm working on correcting an tilted object to its vertically placed position. Below is my drawing illustrating my situation: http://i.stack.imgur.com/0XotT.png Assuming: I have a stick stood on a ...
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2answers
32 views

Decomposing rotation into rotation around certain axis and remaining rotation

Let R be a rotation matrix in three-dimensional euclidean space, R ∈ SO(3). Let v be a unit vector in said space. Is it possible to decompose R into matrices A and B so that following holds? AB ...
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An equivalent definition of the rotation number of a circle homeomorphism

Let $f : \mathbb S^1 \to \mathbb S^1$ be an orientation-preserving homeomorphism. The classical definition of the rotation number is the following: we lift $f$ to a homeomorphism $F : \mathbb R \to ...
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By or through for a rotation

"Rotated through pi rad" vs "Rotated by pi rad" I have heard both used and also heard mentioned that there was a mathematical difference between the two. Is this true or can they be used ...
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multiplication between a matrix and a givens rotation

I want to multiply a matrix A with a givens rotation G. As a reference to this very important link:Click here, they explained in pages 13 and 14 how this multiplication can be achieved. In this PDF, ...
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1answer
41 views

Turning two rotation groups into one

I need to figure out how to turn two rotation groups, each rotating around Z, X then Y into a single rotation group, so that in one set of rotations I might obtain the same positions for a set of ...
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33 views

4x4 Matrix with homogeneous coordinates

I learn for a linear Algebra exam and I have the task: "What is the $4\times 4$ matrix , a rotation about the $\pi/3$ describes in homogeneous coordinates about the axis? What is the ...
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3answers
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Composition of Rotation and Translation in the Complex Plane — Finding Angle of Rotation and Point

A rotation about the point $1-4i$ is $-30$ degrees followed by a translation by the vector $5+i$. The result is a rotation about a point by some angle. Find them. Using the formula for a rotation in ...
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1answer
39 views

why is representing rotations through quaternions more compact and quicker than using matrices??

According to the wikipedia page on Quaternions: The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. However, I have to ...
2
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1answer
41 views

Matrix exponential of the sum of two skew-symmetric matrices

This is my first message in this site. I'm a mechanical engineer with, amongst others, an interest in inertial navigation. I'm currently reading the book "Principles of GNSS, Inertial and Multisensor ...
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1answer
32 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
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27 views

Identity component of SO(2,1)

I am working on Lie groups, and I have several difficulties to show that the identity component of SO(2,1) is the product of an euclidian rotation fixing a vector X and an hyperbolic rotation in a ...
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1answer
54 views

Skew-symmetric matrix property

This page gives the relation $\left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T}$ where $R$ is a DCM (Direction Cosine Matrix), $\vec{v}$ is the angular velocity vector and ...
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2answers
325 views

How to find the shift that minimizes the difference between two vectors?

I am looking for a efficient way to find the value of k that minimizes $\sum(s_t - b_{t+k})^2$ where $s$ and $b$ are N-dimensional vectors and the values are wrapped around like this: $b_{t+k} := ...
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1answer
114 views

$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?

I've been working on a problem related to the 3x3x3 Rubik's Cube where you allow faces to be turned by $45^\circ$ instead of just the usual $90^\circ$. We know for the standard 3x3x3 the cube is ...
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71 views

Decompose 3D rotation matrix into rotation around x, y and z-axis

I have a rotation matrix R, that produces an arbitrary rotation in a 3D space. I would like to decompose it into 3 rotation matrices Rx, Ry and Rz so I can use and apply only xy in plane rotation ...
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1answer
72 views

Area of overlapping squares

I'm working on a programming project and got to the point where I need to find how much is the blue square overlapping each of the other 9 squares. The squares' sides(including the blue one's) are ...
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Rotation of $y=x(1-x)$ about the $x$-axis

In my calculus book, there was a problem involving the rotation of an area $R$, where $R$ is defined as the area enclosed by the function $y=x(1-x)$ and the coordinate axes, about the $y$-axis. This ...
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2answers
71 views

Get the camera transformation matrix (Camera pose, not view matrix)

Let's say that I have an object and a camera (its representation) in a 3D world coordinate system. I have the camera pose to see the object (rotation matrix and translation (eye position)). If I apply ...
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1answer
49 views

Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...
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1answer
22 views

Number of components needed for 3D rotation

Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9. Is it possible to express a 3D rotation using less than 3 real ...