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

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-1
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1answer
35 views

How to rotate a line in 3d space?

I am trying to figure out direction vectors of the arrowheads of an arrow. Basically I'm given a normalized direction vector ...
3
votes
0answers
19 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
0
votes
1answer
41 views

How do I multiple these matrices together?

As a personal brain exercise, I've recently been trying to work out the math involved with rotating vertices around an arbitrary axis in 3D space. To do so, I've been relying very heavily on the ...
0
votes
3answers
29 views

Rotation of a vector

My linear algebra textbook only explained the rotation of a vector in a counterclockwise direction. I'm just wondering what happens if I rotate a vector in the clockwise direction? Do I solve such ...
1
vote
0answers
9 views

Rotationmatrix for coordinate system

this is my first question with regard to mathematics. So, if I made any mistake in terms of naming and/or convention, please let me know. I'm having a little problem with rotating a coordinate system ...
0
votes
1answer
37 views

Geometry/Programming- Draw An Equilateral Triangle Given One Point And A Desired Rotation

I feel this question has a stronger mathematical basis than strictly computer science. I am currently drawing an equilateral triangle given its center and its radius like so. I would like to ...
0
votes
0answers
13 views

How to nullify one of the axises rotation in a rotation matrix?

Let's say that I've got a matrix with some rotation stored. Now I wan't to somehow make an Y rotation equals 0 (or rather make it equal to the starting moment without rotation). How would I do it? I ...
0
votes
2answers
34 views

How do I calculate the inverse of these matrices?

In learning how to rotate vertices about an arbitrary axis in 3D space, I came across the following matrices, which I need to calculate the inverse of to properly "undo" any rotation caused by them: ...
0
votes
1answer
22 views

rotation matrix to axis angle

from wikipedia the above rotation matrix has a rotation of -74 degrees. What does it mean "around the axis (−1⁄3,2⁄3,2⁄3)"? How can I determine how many degrees is rotated on X axis, Y axis and Z ...
0
votes
0answers
7 views

series of Givens rotations to solve this problem

I have an interesting problem where I'm confused. Could anyone please guide me on this? This is the problem: Suppose $A\in R^{9\times3}$ and that we organize a Givens QR so that the subdiogonal ...
1
vote
1answer
13 views

Rotation of velocity vectors in Cartesian Coordinates

I want to rotate a $(X,Y,Z)$ coordinate-system around it $Z$-axis. For the coordinates this can be done with the rotation matrix: $$ R_Z(\theta)= \begin{pmatrix} cos \theta & -\sin(\theta) & ...
1
vote
1answer
45 views

Axis angle rotation as a differential equation

I am trying to solve the equation $\frac{d \vec{x}(\theta)}{d\theta} = \vec{n} \times \vec{x}(\theta)$ where $\vec{x}(\theta)$ is rotated vector $\vec{x}$ by $\theta$ about (normalized) axis ...
0
votes
0answers
36 views

Quarternions from MPU and circumference of circles

First I should mention that my math skills are super basic. I do not understand formulas but I do understand pseudo code, C, C++, and other programming languages. I've been working on a electronics ...
0
votes
1answer
33 views

Plate trick demonstrating SO(3) not simply connected.

I know there is another question about this, but I didn't find the answer adequate. How exactly does the plate trick (http://en.wikipedia.org/wiki/Plate_trick) show that $SO(3)$ isn't simply ...
0
votes
1answer
15 views

What property of rotation preserves relative position?

simple terminology question here. What is the property of rotation which causes objects maintain their relative positions between each other in space? I don't suppose it's the fact that it's an ...
1
vote
0answers
30 views

How to find the axis of rotation needed to rotate a $ 3d$ vector to another $3d$ vector?

I have two vectors $(a,b,c)$ and $(d,e,f)$. How can I find the axis of rotation needed to rotate the first vector to be parallel to the other vector? Thanks
0
votes
1answer
20 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
0
votes
0answers
17 views

How to move coordinate systems using rotation matrices.

I am having some trouble with this question. I understand that the rotation matrix will be 4x4 and that the first 3 columns will just be $u$, $v$ and $n$ transposed but I dont know what I am ...
0
votes
2answers
89 views

How to rotate a matrix by 45 degrees?

Assume you have a 2D matrix. Ignore the blue squares. The first image represents the initial matrix and the second represents the matrix rotated by 45 degrees. For example, let's consider a ...
0
votes
1answer
24 views

Find vector rotated on 3D plane

I don't have access to a computer so I can't give pictures but I will try to make this easy to visualize. Suppose I have a vector $n$. I will now draw a plane normal to this vector that and have that ...
0
votes
0answers
17 views

Inverse Rotation - Original X / Y values

Really stumped. I'm currently writing a program and I am given a rotated rectangular object on a 2d plane. The object has been rotated about it's center point and I need to find out how to get the ...
0
votes
0answers
8 views

Visible area of rotated plane?

Is there any general correlation between rotation and visible area of a plane? If the plane is perpendicular to my point of view I see exactly 100% of it. If it is rotated about the x-axis by 90° it ...
3
votes
0answers
64 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
0
votes
0answers
8 views

point projection into yx rotated plane

I want to simulate depth in a 2D space, If I have a point P1 I suppose that I need to project that given point P1 into a plane x axis rotated "theta" rads clockwise, to get P1' It seems that P1'.x ...
1
vote
0answers
17 views

Rotating two objects

I have two lines. Both created in this format: Line 1 $$line1 = \left\{ \begin{array}{c} startX, startY \\ endX, endY \end{array} \right\}$$ $$line2 = \left\{ \begin{array}{c} startX, startY \\ endX, ...
1
vote
1answer
29 views

Tesseract projection into $3D$

I found this: The tesseract is a four dimensional cube. It has 16 edge points $v=(a,b,c,d)$, with $a,b,c,d$ either equal to $+1$ or $-1$. Two points are connected, if their distance is $2$. ...
0
votes
0answers
27 views

Converting Euler Angle z-y'-x'' sequence to heading

I'm trying to convert a set of Euler Angles to a heading $(0-360)$ degrees. The Euler Angles use the $\ x-y'-x''$ sequence headings, using $\ \psi, \theta, \phi$ as the rotation angles, respectively. ...
0
votes
2answers
53 views

Rotation of Matrices and their interpretation

Given are now two matrices and I have to discuss what the given functions are doing (geometrically). Maybe you can revise/add the following: given are the matrices $ A = \begin{pmatrix} \cos(a) ...
2
votes
1answer
36 views

Obtaining rotation matrix from Euler angles if all three rotations happen at once. Does order of multiplication matter?

I'm having a problem getting my head around Euler Angles. Specifically if I wish to obtain a rotation matrix for a system where pitch, roll and yaw have all changed at once by various values... how ...
1
vote
0answers
47 views

Find the linear (vertical) acceleration using a three axis accelerometer.

I genuinely apologise for what may be a poorly worded question. I'm extremely tired but have a ridiculous huge and important project due in on Monday for my degree. Thank you in advance for any help ...
2
votes
2answers
52 views

Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
7
votes
4answers
142 views

Geometry of the Cayley Transform

I'm trying to understand the geometry of the Cayley transform. Suppose I have a $3 \times 3$ rotation matrix $R$ (i.e an orthogonal matrix with determinant equal to $1$). Let's ignore the corner case ...
2
votes
3answers
75 views

Product of reflections is a rotation, by elementary vector methods

Let $\mathbf{u}$ and $\mathbf{v}$ be two 3D unit vectors. The transform that performs reflection in the plane normal to $\mathbf{u}$ is given by $$ T_{\mathbf{u}}(\mathbf{x}) = \mathbf{x} - ...
0
votes
1answer
23 views

How can I get a rotation angle from a 2d vector?

I have a 2d vector (x,y). And I'd like to obtain from it a rotation angle. For example: I would have 0° degree when (x = positive, y = 0), more than 0° degree when (x = positive, y = positive), and ...
0
votes
1answer
28 views

Describing Cartesian transformations in Cylindrical Polar Coordinates

I have a question about converting functions defined in Cartesian coordinates to a cylindrical polar system. The particular coordinate transformation that I'm reading about is: \begin{equation} ...
0
votes
1answer
14 views

Proving the existence of a ____ number of conditions under a rotation of coordinates.

I'm reading a section on Rotation of Coordinate Systems and this is throwing me off: 'In n-dimensional space, the rotation matrix will have $n^2$ elements, upon which orthogonality relations place ...
0
votes
0answers
28 views

Rotating the spectrum of a bounded operator

If $T$ is a bounded operator on a Banach space $X$, and $\sigma(T)$ is its spectrum, what would be an operator whose spectrum is $\sigma(T)$ rotated by $\theta$? For example, $-T$ has as spectrum ...
1
vote
2answers
15 views

Rotation invariance of higher than 2 dimensions

According to this $f_2(x_1,x_2) = x_1^2 + x_2^2$ is invariant under rotation. I wanted to ask if a function $f_n(x_1,x_2,...,x_n) = x_1^2 + x_2^2 + ...+ x_n^2$ is also rotation invariant. In other ...
0
votes
0answers
33 views

Breaking down an axis-angle rotation into x, y, and z axis component rotations

I have an axis-angle rotation in the form of $(x,y,z,w)$ and I am trying to break that down into the 3 component rotations of $(1,0,0,w_x)$ and $(0,1,0,w_y)$ and $(0,0,1,w_z)$. Is this doable? Any ...
0
votes
2answers
227 views

Getting Euler (Tait-Bryan) Angles from Quaternion representation

Apologies if this has already been answered, but I haven't been able to get a clear answer from looking on Stack Exchange so-far. I'm trying to solve a camera stabilization problem. I have a 2-axis ...
0
votes
1answer
65 views

Rotation formalisms in three dimensions

I'm little bit confused. The Rotations are described by various means Direction Cosines Matrix (DCM); Euler Angles; Euler Axis/Angle; Quaternion. What is the difference between them. How I can ...
1
vote
1answer
68 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
0
votes
1answer
278 views

Quaternion Decomposition

I'm having trouble decomposing a unit quaternion into euler angles (or roll, pitch and yaw). The overall goal is to tell how a phone is rotated with respect to the world. I'm given a unit quaternion ...
1
vote
0answers
44 views

How to determine yaw-pitch-roll orientation by specifying a plane via 3 points?

[Note, this question is an attempt at rephrasing the one posted here, as it has not garnered any attention, unfortunately] Hello, Let's say you have three points in 3D space: A, B and C. Together, ...
0
votes
1answer
17 views

Biggest angle of matrix rotation

How do you find the biggest angle of a matrix rotation? Given an $n \times n$ orthogonal matrix $A$, how do you find $\max_{y \in \mathbb{R}^n, ||y|| = 1} ||y - Ay||_2$? Is there a standard name for ...
0
votes
2answers
24 views

When I use rotation matrix z- position doesn't move. why?

I'm trying to rotate 3 points around x- axis. and three points are (1.33, 2.49, 0),(2.5, 4.33, 0),(0, 5, 0). which is on the xy plane rotation angle is pi/4. then, I followed this cacultation. ...
0
votes
1answer
43 views

Why this 3D rotation matrix doesn't work?

I'm trying to rotate those three red points around x axis about pi/4. and I used this rotate matrix from WiKiPedia. rotation matrix = [[ 1 0 0 ], [ 0 ...
0
votes
2answers
78 views

Transformation Matrix for rotation around a point that is not the origin

I need to find the matrix that rotates an arbitrary point around $\begin{bmatrix}5 \\6\end{bmatrix}$ by 35* anticlockwise. I figure I need to first move the plane to centre it at the origin, perform ...
0
votes
1answer
198 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
0
votes
1answer
135 views

How to modify a transformation that is based on yaw-pitch-roll or phi-theta-psi?

I’m building a model in a 3D simulation program (MSC Adams) and part of that model is a triangular platform which can translate and rotate in the virtual world, as shown in the 2 images below: ...