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

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Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
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0answers
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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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0answers
31 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
263 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
88 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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0answers
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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
50 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
29 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
46 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 ...
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0answers
39 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
20 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 ...
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2answers
21 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
10 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 ...
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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 ...
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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 ...
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1answer
16 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
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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
29 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 $f(x)=x^2$
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0answers
43 views

View change with 4x4 matrix operations?

I have a $\mathbf{S_1}$ as the origin and $\mathbf{S_2}$ as a future origin. $ \mathbf{S_1} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 ...
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1answer
41 views

Implications of $RR^T =\mathbf1$

Let $R:I→SO(3)$, smooth. We know that, for any value of $t∈I$, $R(t)R(t)^T=\mathbf1$, where $\mathbf1$ is the identity matrix. Then, differentiating both sides one finds that ...
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1answer
93 views

Explicit fractional linear transformations which rotate the Riemann sphere about the $x$-axis

Background: Let $S^2$ denote the unit sphere in $\mathbb{R}^3$. By "stereographic projection", I mean the mapping from $S^2$ (remove the north pole) to the complex plane which sends \begin{align*} ...
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0answers
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How could the rotation of an avatars head and body be determined by a single input value?

In a virtual reallity environment I want to rotate a head according to the rotation value provided by the camera sensor of a head mounted display. When the head reaches the boundary of its maximum ...
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1answer
49 views

Oriented surface - rotation of vector field.

Let S be a piecewise smooth oriented surface in $R^3$ with positive oriented piecewise smooth boundary curve $\Gamma:=\partial S$ and $\Gamma : X=\gamma(t), t\in [a,b]$ a rectifiable ...
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2answers
21 views

Rotation Around the Y-Axis

I have an equation: $y = -0.0122625x^2 + 120.38736$ and I want to rotate this around the y-axis and find the volume from the range 0 to 99. I have no idea how to do this and would greatly appreciate ...
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0answers
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How to use 2D Translation and Rotational error to get offset value for new point?

Here I am trying to detect FIDUCIAL points on PCB in real time using camera. After googling for Two days and reading many post and blog. I found that I have to do something called translational error ...
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0answers
10 views

Transform Coordinate system

I would like some help to understand a specific transformation for a coordinate system change as I am not sure about it. I got some sample code so I can see how it is calculated but dont understand ...
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1answer
30 views

Recurrent points and rotation number

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
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Symmetry in recursively defined periodic functions

Say i have some periodic function $f(t)$ that returns a vector ( $R^3$). For each $t$, i can create a unit circle that is defined by the function's direction, that is, $\frac{df(t)}{dt}$. This unit ...
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3answers
38 views

Calculate the position of rocket acted upon simultaneously by multiple thrusters?

I'm looking for an equation that will let me predict the position of a rocket after a period of time given that it is acted upon by multiple forces. By multiple forces I mean the main thruster force ...
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5answers
577 views

Does rotation of a rectangle keep it rectangular?

If I rotate a rectangle by 45°, does it stay rectangular or become something else? I mean do 90° angles stay 90°? I am asking this question because I have some results where the rotated rectangle ...
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1answer
21 views

torus in $SU(2)$ yields a torus in $SO(3)$

in John Stillwell's book "Naive Lie Theory" there is an exercise to explain why a torus in $SU(2)$, (sub group that is isomorphic to $S^1 \times S^1$) yields a torus in $SO(3)$ (in order to prove that ...
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1answer
13 views

Revolving an unknown equation around the x and y axes

The first quadrant region enclosed by the x-axis and the graph of y = ax - x^2 traces out a solid of the same volume whether it is rotated about the x-axis or the y-axis. What is the value of a?
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2answers
50 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
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1answer
22 views

matrix transformation - eigenvector

I am trying to understand eigenvectors. An Eigenvector is nothing more than a vector that points to some place. This pointing vector will then be invariant under linear transformations. Now my ...
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1answer
30 views

understanding quaternions - spatial rotations

I would like to know if my understanding about quaternions is correct please: lets say you have a vector in 3d space. You could rotate the x,y and-z frame on a fixed point so that it is parallel with ...
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2answers
38 views

rotation matrix and vector - understand step calculation

I have an extremely equation, but I just don't understand which step they made to get to the last line. ${\bf W}$ and ${\bf V}$ are all 3d vectors. A is a rotation matrix. How did they get that ...
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2answers
68 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
2
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0answers
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Free groups of rotations of the sphere

Is the following conjecture true: If $G$ is a group of rotations of the sphere and $G$ contains two noncommuting rotations of infinite order, then $G$ has a free subgroup of rank $2$. By the Tits ...
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0answers
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How can I define the order of rotation from one rotation matrix to another?

I am trying to rotate a koordinatesystem in the defined order of z-x-y. I have the matrix $rot_x$ describing the rotation around the x-Axis, the matrix $rot_y$ describing the rotation around the ...
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1answer
37 views

how to rotate scaled-vector (orientation) by scaled-vector (rotation)

Recently I got the physics-engine portion of my 3D simulation / game engine working correctly. The most convenient way to store and compute position and orientation are in 3-element vectors (though my ...
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1answer
62 views

free groups of rotations

The question of which pairs of rotations of the sphere are independent goes back to Hausdorff, who produced such a pair a century ago. "Independent" means "are free generators of a free group". The ...