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

learn more… | top users | synonyms

0
votes
2answers
20 views

Difference between quaternions depends on initial rotation

The difference $\Delta q$ between two quaternions $q1$ and $q2$ can be calculated as $\Delta q = q1\cdot q2^{-1}$, where $^{-1}$ is the quaternion conjugate. When numerically evaluating the ...
1
vote
1answer
25 views

Rotation about z axis using quaternions

I am working with quaternions and rotation, but I am missing something about how a rotation expressed as a quaternion works. I also discovered that there are different convention for quaternions (JPL, ...
0
votes
2answers
52 views

How do I rotate a vector 90 degrees in a random direction?

I'm building a tree generator and I'm at the point where I want to have sub branches branch off at right angles off the current branch, in random directions. I have a 3D vector defining the direction ...
0
votes
2answers
36 views

Quaternion angle - Opengl rendering

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). I am trying to calculate the angle of rotation around all the three axes and Render a 3D cube using opengl to immitate the ...
0
votes
1answer
14 views

Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
1
vote
0answers
20 views

Finding regular values of $f(A)=A^2$

So I am trying to solve a few problems in differential geometry I found on the U of C website, and got stuck on this one: What are the regular values of the map $f: SO(3) \rightarrow SO(3)$ given by ...
0
votes
2answers
37 views

$2x^{1/4}$ rotated around $y = 2x$

This is the question: find the volume created by rotating $2 x^{1/4}$ around $y=2x$. I was able to define the distance between the two lines as $y/2 - (y/2 )^4$. However, I can't find the radius that ...
1
vote
0answers
10 views

Show that $δ_{KL}$ is a Cartesian tensor

By using the definition of the Kronecker delta $δ_{KL}$, show that $δ_{KL}$ is a Cartesian tensor, that is $δ'_{MN} = L_{MK}L_{NL}δ_{KL}$ under the rotation $X_K = L_{MK}X'_M$. Solution: Using the ...
0
votes
0answers
10 views

In a transformation matrix, why is $Y$-axis ($-\sin$) in left most column as opposed to right like X and Z [duplicate]

$4 \times 4$ Transform Matrix with axis columns $XYZ$ left to right $X$-axis rotation: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & \cos & -\sin & 0 \\ 0 & \sin & \cos ...
0
votes
0answers
37 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
0
votes
2answers
44 views

Prove that 3d rotation is linear

In a 2d space, a transformation is linear if $f(v+w) = f(v) + f(w)$ and $f(kv) = k*f(v)$, and rotation preserves addition so it is linear. In a 3d space, similar rules apply: $(x, y, z) + (l, j, k) = ...
0
votes
1answer
7 views

Estimate angular velocity and angular acceleration from a point cloud sequence

Lets say I have the a set of points $P = \{p_1, p_2, ...\}, p_i \in R^3$ that change position with time. These points are part of a rigid body and I record these positions in order to estimate its ...
0
votes
0answers
66 views

Rotating one 3d-vector to another only by using rotations about the coordinate axes.

If I have a vector v=(x,y,z) and would like to transform another vector u by using only rotations about the coordinate axes to be in the direction of v, how can I find required angles and the order of ...
0
votes
1answer
34 views

How to decompose a unit quaternion into 3 Tait-Bryan quaternions instead of 3 real numbers?

I'm familiar with the formulas for decomposing a unit quaternion $Q$ into chained Tait-Bryan angles $\phi\theta\psi$ (Wikipedia has the formulas for the $zyx$ chain here), but I'm looking to instead ...
0
votes
0answers
29 views

Align 2 rotated rectangles [closed]

enter image description here Given a rectangle formed by points A, B, C, D (blue), i can calculate all the following points: E, F, G, H, I, L, M, N, O. 1) So with the following Values: A, B, C, D ...
0
votes
1answer
15 views

Rotations about the origin

Let R(θ) denote a rotation matrix which rotates a point $x$ in $S^2$ anticlockwise about the origin through a given angle θ. (Where $S$ is the set of real numbers) How do you illustrate that this ...
-1
votes
0answers
12 views

Create 3d rotation matrix given spherical coordinates? [closed]

Is it possible to form a 3d rotation matrix given a spherical coordinates $\theta$ and $\phi$ ?
0
votes
2answers
65 views

Calculate Rotation Matrix to align k n dimensional vectors

I have a $k$ number of $n$-dimensional vectors written with respect to two rotated frames: $X= \{\vec{x}_1,\vec{x}_2,...,\vec{x}_k\}$ and the same rotated vectors: $X'= ...
1
vote
1answer
60 views

Why does rotation by a quaternion require multiplying two times?

Given a vector $p$, to rotate it by a quaternion $q$, we use the formula: $$p' = q p \hat{q}$$ where $\hat{q}$ is the conjugate of $q$. But if we use rotational matrices, then it's just $$p' = ...
0
votes
0answers
26 views

Convert a 3d vector into a rotation matrix?

Is it possible to compute a Rotation matrix given a 3d vector given in the Euclidean space? and if not what would it need? An illustration of my situation. Illustration of my problem I have a ...
0
votes
0answers
25 views

How do i compute how much i can rotate my tool?

I am at moment trying to implement an Ball tracker for a robot arm with a stereo camera monted on it as its tool. Illustration: http://m.imgur.com/5oojXdh The camera provide me with an dx, dy, dz ...
0
votes
0answers
15 views

How do i convert an x,y,z to an Q configuration?

I am trying to implement a tracking application for a robot arm, which purpose is relocate itself based on the position of an object seen from the tool point. illustration: http://imgur.com/5oojXdh ...
0
votes
1answer
19 views

Rotations around an axis

I am given a linear transformation $T:\mathbb{R^3} \rightarrow \mathbb{R^3}$ The transformation is linear and is defined by taking a vector in $\mathbb{R^3}$ and rotating it around the axis ...
0
votes
1answer
19 views

Applying Rotation Matrix to Rotate Point (2,4,4) wrt y onto X-Y Plane

My linear algebra is a little rusty. Given an arbitrary point P(x,y,z), how do I determine the theta needed to apply a rotation matrix that will rotate the point onto the X-Y plane, with respect to Y? ...
0
votes
0answers
24 views

Graphing calculator leaving gaps in a drawn graph of a rotated parabola

I am graphing the equation of two rotated parabola on the graphing calculator and, after finding the y= form for each using the quadratic formula and entering them into a program to graph them they ...
1
vote
1answer
31 views

Calculating a quaternion that represents a given rotation

This is the first time I'm attempting to do a quaternion and I am not quite getting the concept. This is part of a 3 calculation homework question The initial question is Given a 3-D point at ...
-1
votes
2answers
26 views

Proof there isn't a vector u such Su=u where S is the rotation transformation in R2 [closed]

We have the rotation matrix \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{pmatrix} Proof: there isn't exist a vector $u \in\ {\mathbb{R}^2}$ ($u\neq0$) such ...
0
votes
1answer
26 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert ...
0
votes
0answers
9 views

How do we rotate a vector in a 3D cartesian plane with the plane of rotation specified?

I know the rotation method of a complex number that : "If we multiply a complex number $x + \iota y$ by $e^{\iota \theta}$ where $\theta$ is the angle of rotation in anticlockwise sense then we get a ...
-4
votes
2answers
36 views

If $A$ is a rotation matrix by $\theta$, then what does $A^T$ do?

Little help here? If multiplication by $A$ rotates a vector $x$ in the $x$-$y$ plane through an angle $\theta$, then what is the effect of multiplying $x$ by $A^T$. Explain your reasoning. Any help ...
0
votes
0answers
19 views

How to calculate coordinates $(x,y)$ of rotated polygon?

I have coordinates $x,y$ of a point (red point on the image). If I rotate the image with a specific angle (for example 30 degrees) how can I get the coordinates $x,y$ in the new polygon (which is ...
2
votes
1answer
20 views

3D calculate new location of point after rotation around origin

I've tried to boil down my problem as much as possible. I've got two questions, but really I'd be satisfied enough just knowing how to accomplish the first one. I'm looking to do this programatically, ...
0
votes
0answers
13 views

Rotate the plane through 90 degree?

I have a problem from the book "Linear Algebra and Its Applications" by G. Strang. What $3$ by $3$ matrix represents the transformation that rotate the $x$-$y$ plane through $90^{\circ}$, ...
1
vote
2answers
13 views

Robustly map rotation matrix to axis-angle

The Wikipedia article for rotation matrix gives the following formula for converting from rotation matrix, $Q$, to axis-angle, $u$ and $\theta$: $$ \begin{align} x &= Q_{zy} - Q_{yz} \\ y &= ...
0
votes
1answer
22 views

Why 90 degree rotation form a special case in Euler angle representation of rotation

I was reading a document illustrating euler angles representation of rotations. A Rotation that depends on 3 angles $\alpha$ around z axis, $\beta$ around y axis, and $\gamma$ around x axis is ...
11
votes
2answers
61 views

Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
0
votes
1answer
27 views

Eigenvalues/Eigenvectors of a Rotational Matrix

Sorry for boring you my friends. I am haunted by a demonstration in the book. Here is the link: http://robotics.caltech.edu/~jwb/courses/ME115/handouts/rotation.pdf The question is mainly about the ...
0
votes
0answers
18 views

Applying rotation matrix on inclined plane

I want to rotate an inclined plane to a flat surface. I think I can use the Euler angles to perform this operation. Using following points (Matlab): ...
0
votes
1answer
25 views

2D coordinates of rotating a “bent line”?

I have this problem, when I am given a point A an an XY plane, and I need to find the coordinates of a point B that is of a constant distance of my point A, and my OAB angle is fixed (O being the ...
0
votes
0answers
39 views

I'm looking for a rotation matrix for following transformation

I'm working with a 3D camera and I found out the formula to transform the camera measurements to real world coordinate system when you have a rotation around x and y (no z rotation). ...
0
votes
1answer
26 views

What is the rotation index of a figure 8?

Is it 0 since the total turning angle covers one clockwise circle and one counterclockwise circle thus making the total 0 and the rotation index 0?
0
votes
0answers
35 views

Calculate ψ knowing object orientation in 3D through forward and up vector

I've got a so called right, up, forward tridimensional reference plane and an object $P$ in it. Its orientation in space is defined by two vectors, forward and up: -forward gives azimuth $θ$ and ...
0
votes
1answer
26 views

Concatenating two Rotation-Matrices

I have two $2\mathrm{D}$-planes in $3\mathrm{D}$-space with orientation parameters expressed as rotation $R_1$ and translation $T_1$ and rotation $R_2$ and translation $T_2$ with respect to some ...
0
votes
1answer
22 views

How to find the amount of degrees to rotate a vector to be 90 degrees another vector?

I have a vector V that rotates around an axis K and a vector N all in 3D space. I need to find how much to rotate the vector V around S so that it lies 90 degrees to N. So far I have been doing it ...
0
votes
0answers
25 views

Finding angle between y axis in two rotated coordinate systems

I basically have two coordinate systems that have the same origin, and can measure the coordinates of a vector (but only one) in respect to both of them. I need to calculate the angle between the y ...
3
votes
1answer
27 views

Rotation of a matrix

Sorry for boring you my friends before the holiday. I am haunted by a question of rotation of a matrix. Suppose that we have a special matrix $\Omega$ takes the form of: $\Omega = \left[ \begin ...
8
votes
4answers
183 views

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2.

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2. Here's my attempt at a solution: We know that any rotation in $\mathbb{R^3}$ can be ...
1
vote
0answers
28 views

Unifying Transformations in Complex 3-Space

I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the ...
2
votes
1answer
27 views

Normalizing a quaternion

How do I normalize a quaternion $$q=w + \mathbf ix + \mathbf jy + \mathbf kz = a + v$$ ? I already know: The normalized quaternion is called unit quaternion and can be calculated in this way: $$U_q = ...
0
votes
0answers
16 views

Rotations on Bloch Sphere

I am trying to convince myself that the operators $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$, are indeed the rotation operators on the Bloch sphere. Lets say we have a state vector $$|\psi \rangle = ...