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

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2
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1answer
19 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
votes
0answers
36 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 $y_2=(x_1-x_2)$. ...
1
vote
1answer
18 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 ...
0
votes
0answers
18 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
votes
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 ...
0
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0answers
9 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 ...
0
votes
0answers
13 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 ...
0
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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
votes
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
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 $\mu(R_f)$ [$=$ ...
0
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0answers
26 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 ...
0
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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 ...
1
vote
3answers
42 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 $f(x)=x^2$
-1
votes
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 ...
1
vote
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 ...
3
votes
1answer
91 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*} ...
1
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0answers
14 views

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 ...
-1
votes
1answer
48 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 ...
1
vote
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 ...
1
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0answers
11 views

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 ...
0
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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 ...
0
votes
1answer
29 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? ...
0
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0answers
7 views

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 ...
0
votes
3answers
37 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 ...
2
votes
5answers
570 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 ...
1
vote
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 ...
0
votes
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?
1
vote
2answers
49 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
2answers
67 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
votes
0answers
39 views

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 ...
0
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0answers
16 views

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 ...
0
votes
1answer
36 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 ...
3
votes
1answer
61 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 ...
1
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0answers
83 views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
2
votes
0answers
39 views

Calculate new pitch and roll after rotating about the z axis

I am wanting to know how to find out the new pitch and roll values when rotating around a circle. I have become a little stuck on how to achieve this, but hopefully someone will be able to point me in ...
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0answers
33 views

Transformation matrix for two rotations

If I am required to compute the full transformation matrix compromising of the following sequence of operations: rotation by $30$ degrees about $x$-axis translation by $1$, $-1$, $4$ in $x$, $y$ and ...
1
vote
1answer
47 views

3D rotation of an object with respect to another object's rotation

I am writing a python code to translate and rotate an object with respect to another object. Please take a look at the picture bellow: The smiley face and the arrow have initial poses (position ...
3
votes
3answers
61 views

How many degrees of freedom would a rotation matrix in R5 have?

I understand that a rotation matrix in R3 has three degrees of freedom because there is three linearly independent planes that the rotation can take place in. How does this translate to R5?
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0answers
68 views

How do I get the rotation between two rotationmatrices?

I am stuck on a little rotation problem. The problem: I have 2 rotation matrices $A$ and $B$. $A$ and $B$ are relative to the coordinate system O. $A$ and $B$ are Quaternion rotation matrices. I am ...
1
vote
1answer
16 views

Counterclockwise rotation matrix

If I take the basis $(\vec{e_x},\vec{e_y})$ and make a rotation counterclockwise of angle $\theta$, I end up with two new vectors $(\vec{u},\vec{v})$ such that : $\vec{u} = \cos\theta \vec{e_x} + ...
1
vote
0answers
38 views

rotation matrix - satellite control system

I am in charge with developping a satellie control system. There is something that I either understood incorrectly or either is wrong in the explanation. I am checking if I am able to get the values ...
1
vote
0answers
11 views

two rotations of line segment in different order

there are given are two rotations: $R_1$ and $R_2$, and a line segment $AB$. the image of $R_1R_2(AB)$ is a line segment that is parallel to the line segment $R_2R_1(AB)$. my kindly request is to the ...
0
votes
0answers
18 views

Education tool for learning 3D angles

I hope it is not an off-topic. I have started working on 3D frame transformation and transforming a vector such as acceleration or angular velocity from one coordination to earth coordination. My ...