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Questions tagged [rotations]

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

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192 votes
21 answers
385k views

Calculate Rotation Matrix to align Vector $A$ to Vector $B$ in $3D$?

I have one triangle in $3D$ space that I am tracking in a simulation. Between time steps I have the previous normal of the triangle and the current normal of the triangle along with both the current ...
user1084113's user avatar
  • 2,099
101 votes
9 answers
46k views

Why is the product of two rotation matrices not commutative?

Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?
Navin Prashath's user avatar
94 votes
3 answers
5k views

Modelling the "Moving Sofa"

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
newzad's user avatar
  • 4,865
91 votes
7 answers
52k views

Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix

The rotation matrix $$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\...
Alf's user avatar
  • 2,597
84 votes
5 answers
23k views

Math behind rotation in MS Paint

For those who don't know, MS Paint only has the options to rotate an image by right angles. To carry out an arbitrary rotation ($\theta^\circ$), the following hack is suggested: Horizontal skew ...
kuch nahi's user avatar
  • 6,799
63 votes
5 answers
177k views

How do you rotate a vector by a unit quaternion?

Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
Narf the Mouse's user avatar
61 votes
15 answers
47k views

Area of a square inside a square created by connecting point-opposite midpoint

Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each. $H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively. What will the area of the square formed in the middle be? I know that ...
Agile_Eagle's user avatar
  • 2,952
51 votes
3 answers
104k views

Rotation Matrix of rotation around a point other than the origin

In homogeneous coordinates, a rotation matrix around the origin can be described as $R = \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0\\\sin(\theta) & \cos(\theta) & 0 \\ 0&0&...
Dschoni's user avatar
  • 838
45 votes
4 answers
230k views

Find the coordinates of a point on a circle

I have a circle like so Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and 360 degrees. For example, I have a radius r ...
CoderTheTyler's user avatar
43 votes
1 answer
31k views

Generalized rotation matrix in N dimensional space around N-2 unit vector

There is a 2d rotation matrix around point $(0, 0)$ with angle $\theta$. $$ \left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right] $$ Next, ...
Ivan Kochurkin's user avatar
41 votes
11 answers
33k views

"Random" generation of rotation matrices

For a current project, I need to generate several $3\times 3$ rotation matrices for input into an algorithm. I thought I might go about this by randomly generating the number of elements needed to ...
bob.sacamento's user avatar
40 votes
2 answers
130k views

Understanding rotation matrices

How does $ {\sqrt 2 \over 2} = \cos (45^\circ)$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? ...
user3871's user avatar
  • 679
33 votes
3 answers
24k views

Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all that I learned about orthogonal matrix is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my surprise, I ...
Alby's user avatar
  • 575
32 votes
4 answers
22k views

Can rotations in 4D be given an explicit matrix form?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is ...
user263007's user avatar
30 votes
3 answers
88k views

How to find an angle in range(0, 360) between 2 vectors?

I know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the ...
Savail's user avatar
  • 455
29 votes
9 answers
4k views

Why do complex numbers lend themselves to rotation?

In the introductory complex analysis course I am taking, nearly every theorem relates to rotation and argument. Why do complex numbers love doing this so much? I can understand why these theorems work;...
Christopher Lee's user avatar
29 votes
3 answers
50k views

How to rotate one vector about another?

Brief Given 2 non-parallel vectors: a and b, is there any way by which I may rotate a about b such that b acts as the axis about which a is rotating? Question Given: vector a and b To find: vector ...
reubenjohn's user avatar
29 votes
2 answers
87k views

Are Euler angles the same as pitch, roll and yaw?

I am wondering if pitch, roll and yaw are used to represent Euler angles? If not, what's the relationship between them? From wiki, I know that Euler angles are used to represent the rotation from ...
Ovilia's user avatar
  • 393
28 votes
3 answers
61k views

Rotation matrix in spherical coordinates

When given arbitrary point on a unit sphere $a = (\theta, \phi)$ and an arbitrary axis $\vec{A}=(\Theta, \Phi)$, can we have an algebraic expression for $a_1=(\theta_1, \phi_1)$ which is a rotation of ...
Andrew's user avatar
  • 381
27 votes
9 answers
4k views

What does it mean to represent a number in term of a $2\times2$ matrix?

Today my friend showed me that the imaginary number can be represented in term of a matrix $$i = \pmatrix{0&-1\\1&0}$$ This was very very confusing for me because I have never thought of it ...
Olórin's user avatar
  • 5,455
26 votes
12 answers
4k views

Stuck on a Geometry Problem

$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$. Determine the length of ...
blackened's user avatar
  • 1,115
26 votes
5 answers
20k views

Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
user76284's user avatar
  • 5,997
24 votes
2 answers
5k views

What is the exact and precise definition of an ANGLE?

On wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of ...
user103816's user avatar
  • 3,859
22 votes
7 answers
18k views

Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?

I know if $A^TA=I$, $A$ is an orthogonal matrix. Orthogonal matrices also contain two different types: if $\det A=1$, $A$ is a rotation matrix; if $\det A=-1$, $A$ is a reflection matrix. My question ...
Shiyu's user avatar
  • 5,228
22 votes
1 answer
1k views

Maximum angle between a vector $x$ and its linear transformation $A x$

Let $A \in \mathbb{R}^{n \times n}$ be a given symmetric positive definite matrix. I would like to find the maximal rotation $A$ can create over any unit vector $x \in \mathbb{R}^n$. In other words, ...
karakusc's user avatar
  • 1,512
21 votes
4 answers
15k views

Rotational invariance of cross product

I'm looking for a proof that $$ ( Ra \times Rb ) = R ( a \times b ) $$ where $\times$ is the three-dimensional cross product, and $R$ is a rotational matrix (such that $\det R = 1$ and $R^T R = I$). ...
Howdy Ho's user avatar
  • 655
20 votes
7 answers
26k views

Rotating one 3d-vector to another

I have written an algorithm for solving the following problem: Given two 3d-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$. However, I am not sure if the following ...
Libor's user avatar
  • 1,305
19 votes
4 answers
132k views

Rotating x,y points 45 degrees

I have a two dimensional data set that I would like to rotate 45 degrees such that a 45 degree line from the points (0,0 and 10,10) becomes the x-axis. For example, the x,y points ...
user1728853's user avatar
19 votes
3 answers
5k views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
JCW's user avatar
  • 678
19 votes
7 answers
15k views

Jacobian matrix of the Rodrigues' formula (exponential map)

I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational. ...
nils's user avatar
  • 333
19 votes
5 answers
14k views

Averaging quaternions

Given multiple quaternions representing orientations, and I want to average them. Each one has a different weight, and they all sum up to one. How can I get the average of them? Simple multiplication ...
Hannesh's user avatar
  • 725
19 votes
3 answers
7k views

Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
user avatar
18 votes
6 answers
2k views

Why can't rotations be represented by purely imaginary quaternions?

I imagine this question has a straightforward answer, but I haven't been able to think of it on my own. It's well-accepted that you can represent a rotation in three-space with a unit quaternion. In ...
Draconis's user avatar
  • 1,453
18 votes
1 answer
50k views

How to convert Euler angles to Quaternions and get the same Euler angles back from Quaternions?

I am rotating n 3D shape using Euler angles in the order of XYZ meaning that the object is first rotated along the X axis, then Y...
Amir's user avatar
  • 435
17 votes
7 answers
18k views

Finding the rotation matrix in n-dimensions

Suppose that we know two real vectors with n components, which are linked by some arbitrary transformation/scaling/rotation/shearing... Now, I think that it is possible to know which is the scaling ...
L. B.'s user avatar
  • 303
17 votes
1 answer
41k views

What is the parametric equation of a rotated Ellipse (given the angle of rotation)

The Formula of a ROTATED Ellipse is: $$\dfrac {((X-C_x)\cos(\theta)+(Y-C_y)\sin(\theta))^2}{(R_x)^2}+\dfrac{((X-C_x) \sin(\theta)-(Y-C_y) \cos(\theta))^2}{(R_y)^2}=1$$ There: - $(C_x, C_y)$ is the ...
Gil Epshtain's user avatar
17 votes
2 answers
13k views

Proof of the extrinsic to intrinsic rotation transform

Wikipedia states that: Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice-versa. For instance, the ...
Nathaniel Bubis's user avatar
17 votes
1 answer
1k 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 ...
Brandon Enright's user avatar
16 votes
3 answers
24k views

Comparing two rotation matrices

Problem I want to compare two rotation matrices $R_A$ and $R_B$ both representing the orientation of the same point cloud in space, but computed from different methods. The idea is to have an ...
JonasVautherin's user avatar
16 votes
6 answers
21k views

Integrating body angular velocity

I've been reading over some very comprehensive notes on attitude representation, which were compiled by James Diebel, a Stanford student: http://www.swarthmore.edu/NatSci/mzucker1/e27/...
staple's user avatar
  • 355
15 votes
3 answers
5k views

Commutative Rotations

In three dimensions, I know that in general rotations on the unit sphere are non-commutative, but I was wondering if there is a subset/subgroup of rotations that are commutative, and what this type of ...
kηives's user avatar
  • 787
15 votes
2 answers
17k views

tensor rotation

why does tensor rotation require multiplication by the rotation matrix twice, once from the right and once from the left by the inverse? if $T$ is the tensor I wish to rotate and $R$ is the rotation ...
Rubenz's user avatar
  • 421
15 votes
5 answers
474 views

Can an arbitrary 3-d shape be fitted inside a cube so it touches all the sides?

In 2-d space, it is possible to take any shape and fit it inside a square such that it touches all the sides of the square. In other words, its projection on the x-axis is the same as its projection ...
Rohit Pandey's user avatar
  • 6,893
15 votes
1 answer
18k 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 ...
Jan's user avatar
  • 261
14 votes
2 answers
49k views

Inversion of rotation matrix

For example, I have a two-dimensional rotation matrix $$ \begin{bmatrix} 0.5091 & -0.8607 \\ 0.8607 & \phantom{-}0.5091 \end{bmatrix} $$ and I have a vector I'd like to ...
aleksv's user avatar
  • 301
14 votes
4 answers
8k views

The difference between applying a rotation matrix to a vector (points) and to a matrix (transformation)

Suppose that the rotation matrix is defined as $\mathbf{R}$. Then in order to rotate a vector and a matrix, the following expressions are, respectively, used $\mathbf{u'}=\mathbf{R} \mathbf{u}$ and ...
KratosMath's user avatar
14 votes
3 answers
5k views

What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + \vec{...
JJW5432's user avatar
  • 387
14 votes
1 answer
437 views

Is my paper on a number system that allows arithmetic on 3D vectors useful?

I have constructed a number system similar to the quaternions, but with three dimensions, not four, ie vectors of the form $(x, y, z)$. It has fairly well-behaved multiplication and division and every ...
Tad Boniecki's user avatar
14 votes
1 answer
664 views

Can any higher-dimensional Spheres be rotated everywhere equally?

You can rotate a circle so that every point on it (just the perimeter, not the interior) moves "equally". That is, every point moves with the same speed and even has the same "acceleration" (first-...
RBarryYoung's user avatar
14 votes
2 answers
5k views

$3D$ rotation matrix uniqueness

Given a $3D$ rotation matrix $R$ in a basis $B$. Can we consider $R$ as being unique in $B$? Is there any other $3D$ rotation matrix $R'$ representing the same $3D$ rotation in $B$? How could I prove ...
Korchkidu's user avatar
  • 317

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