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

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2answers
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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
26 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
15 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
49 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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0answers
2 views

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 ...
1
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0answers
52 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 ...
2
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2answers
36 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 ...
0
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1answer
41 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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0answers
26 views

600-cell rotation [closed]

The more general mathematical issue is about visualizing 4D rotations, and determining how 4D objects will project into 3D using stereographic projection. If we have a 4D polytope, say with a vertex ...
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0answers
18 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
18 views

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 ...
2
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0answers
34 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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0answers
26 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 ...
2
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2answers
28 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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0answers
36 views

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 ...
3
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0answers
25 views

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 ...
0
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0answers
19 views

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, ...
0
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1answer
40 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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0answers
32 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
17 views

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 the ...
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0answers
27 views

Linear Algebra conversion?

I'm trying to create the following matrix via programming but I can't seem to get the equation right (https://upload.wikimedia.org/math/f/b/a/fbaee547c3c65ad3d48112502363378a.png). $$R = ...
1
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1answer
36 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
39 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 ...
0
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1answer
25 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'} = ...
2
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0answers
26 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 ...
0
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1answer
50 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 ...
5
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2answers
316 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} := ...
12
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1answer
100 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 ...
2
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0answers
27 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 ...
1
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1answer
53 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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0answers
18 views

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
52 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 ...
0
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1answer
48 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 ...
2
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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 ...
3
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0answers
40 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
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1answer
24 views

Dot product of of quaternion-rotated vectors

I'm reading http://people.csail.mit.edu/bkph/articles/Quaternions.pdf and it says "it is easy to show that the operation preserves dot-products." on the page 3. But how is it done? I tried to make a ...
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0answers
19 views

Coin rolling, not sliding - König's Theorem

A homogeneous coin of mass M rolls, without sliding, along the x-axis with the axis of rotation being parallel to the z-axis. Let $\Theta$ be moment of inertia regarding that axis and $\vec{V}$ be ...
3
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2answers
23 views

Cylinder inside a cylinder - rotation.

A homogeneous cylinder with radius a and mass m rolls in a hollow cylinder with radius R. Determine the kinetic energy of the cylinder as function of $\dot{\theta}$. Alright, I found this ...
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0answers
13 views

How to transform Tait-Bryan-Angels to different rotation orders?

I am having trouble finding or understanding how to get Tait-Bryan-Angels from a rotationmatrix. I have a given rotation matrix $R_q$ which was calculated from the quaternion $q$. I know how to ...
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0answers
14 views

Direction cosines on a plane

Guys I have two sensors placed on a body, one is on {-x,-y} plane aligned at 30 degrees wrt -x axes, another lies on {+x,-y} plane with 30 degrees from +x axes. For such case should my DCM for 1st ...
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0answers
11 views

How can I get the angle from one Y height to another?

From an entity, I am attempting to get a Direction vector that is pointing directly torwards another entity in 3D space. I have successfully accomplished this by ...
2
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1answer
23 views

The space of orientations of a 3-d object ($SO(3)$, $RP^3$, $S^3$, $S^2 \times S^1$, etc)

It seems like I am missing something really basic here. I am thinking of the following 2 representations of the orientations of a 3d object (excluding reflections). Take a sphere at the origin for ...
0
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2answers
32 views

Transformation matrix: rotation in $\mathbb R^3$

Operator $\phi: \mathbb R^3 \to \mathbb R^3$ is rotation around line $p: x_1 - x_2 = 0,$ $ x_3=0$, $\phi (0,0,2) = (\sqrt2,-\sqrt2,0)$. I need to find transforamtion matrix $A$: $\phi(x)=Ax$ in ...
0
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1answer
18 views

1D FFT on rotated image column by column

I am facing a problem: performing 1D FFT on a rotated column by column on a rotated image, described as following: Original Image: Rotated Image: What I have: original image convolution ...
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2answers
17 views

Surface area of revolution for curves symmetrical on the axis of revolution.

I understand that surface area generated when an curve is rotated on the x axis by 2Pi radians is given by: 2Pi∫yds How is this area affected when the object is symmetrical on the x-axis, e.g. an ...
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0answers
18 views

How to rotate a 2nd derivative gaussian function?

I have a 2nd gaussian derivative in y and a normal gaussian in x, which results in the function: $$ f\left ( x,y \right ) =\frac{- \exp ^{-\left (\frac{x^{2} +y^{2}}{2\cdot \sigma^{2} } \right ...
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0answers
14 views

Solid of revolution in cylindrical coordinates

$f:[a,b]\rightarrow\mathbb{R}$ is continuous and $R_f\subset\mathbb{R}^3$ is the solid of revolution that resulted from the rotation of the graph of $f$ around the x-axis. Evaluate $\mu(R_f)$ [$=$ ...
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0answers
38 views

Calculating two rotation angles from xyz coordinates for dummies

This post is a bit verbose so that others who come later may benefit from my thick headedness. I am attempting to construct a primitives composition and constructed solids geometry parser/processor ...
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0answers
17 views

Rotation Invariant Descriptors for Bivariate Polynomial Surfaces

I start with a simple example. Consider: $$ z = x^2 + y $$ and $$ z = y^2 + x $$ Visually speaking, both of these are essentially the same surfaces rotated by 90 degrees about the z-axis. I am ...
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3answers
43 views

Rotate the graph of a function?

How do I rotate a graph of a function around a point, and show it in the related equation? An example could be $f(x)=\lvert x\rvert$ (absolute Value) and $f(x)=x^2$