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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
384k 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
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
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
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
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
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
2 votes
2 answers
562 views

Consider a right angled $\triangle PQR$ right angled at $P$ i.e ($\angle QPR=90°$) with side $PR=4$ and area$=6$.

Consider a right angled $\triangle PQR$ right angled at $P$ i.e ($\angle QPR=90°$) with side $PR=4$ and area$=6$. If $\triangle PQR$ is rotated through $360°$ about the side $PR$ , what is the $TSA$ ...
pi-π's user avatar
  • 7,426
1 vote
3 answers
3k views

Representing rotations using quaternions

I'm learning Unity and came across a situation where rotations are represented as Quaternions. I've heard that they where used in computer graphics, but never had to use them until now. What I can't ...
moray95's user avatar
  • 1,037
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
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
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
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
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
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
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
  • 1,307
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
11 votes
4 answers
875 views

Simple examples of $3 \times 3$ rotation matrices

I'd like to have some numerically simple examples of $3 \times 3$ rotation matrices that are easy to handle in hand calculations (using only your brain and a pencil). Matrices that contain too many ...
bubba's user avatar
  • 43.5k
10 votes
2 answers
6k views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
Blue bear's user avatar
  • 101
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
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
6 votes
6 answers
6k views

How can I calculate a $4\times 4$ rotation matrix to match a 4d direction vector?

I have two 4d vectors, and need to calculate a $4\times 4$ rotation matrix to point from one to the other. edit - I'm getting an idea of how to do it conceptually: find the plane in which the vectors ...
Reykjavik's user avatar
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
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
11 votes
2 answers
11k views

What are all the elements of the group of symmetries of a regular tetrahedron?

I can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on 'top' and we do to the other $3$ as we did in a ...
user avatar
11 votes
2 answers
8k views

Rotating vector functions

Suppose you have a vector function of space: $\vec{E}(x,y,z)=Ex(x,y,z)\hat{x}+Ey(x,y,z)\hat{y}+Ez(x,y,z)\hat{z}$. Suppose now you want to rotate the whole vector function by using a unitary rotation ...
SDiv's user avatar
  • 2,530
7 votes
1 answer
351 views

Does the notion of "rotation" depend on a choice of metric?

Consider the statement: The Euclidean metric on $\mathbb{R}^n$ is rotationally invariant. I interpret this to mean (is this interpretation correct?): The Euclidean metric on $\mathbb{R}^n$ is ...
Chill2Macht's user avatar
  • 21.2k
6 votes
1 answer
5k views

Expression of rotation matrix from two vectors

What is the matrix expression of the rotation matrix in 3D which turns a vector $\vec{a}$ into a vector $\vec{b}$, with both vectors given by their coordinates? ($\vec{a} = (a_x, a_y, a_z)$ and $\vec{...
tmlen's user avatar
  • 358
5 votes
3 answers
1k views

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \...
Widawensen's user avatar
  • 8,224
4 votes
2 answers
800 views

The relation between axes of 3D rotations

Let's suppose we have two rotations about two different axes represented by vectors $v_1$ and $v_2$: $R_1(v_1, \theta_1)$, $R_2(v_2,\theta_2)$. It's relatively easy to prove that composition of ...
Widawensen's user avatar
  • 8,224
4 votes
1 answer
7k views

Finding rotation axis and angle to align two 3D vector bases

I have asked this question before and, while the accepted answer solved my problem back then, I am still interested in finding the rotation axis and angle. Let me rephrase the problem here: I would ...
HLorenzi's user avatar
  • 175
3 votes
1 answer
7k views

Finding Rotation Axis and Angle to Align Two "Oriented Vectors"

In general, one can align a 3D vector $\vec A$ to another 3D vector $\vec B$ by rotating $\vec A$ around the axis $\| \vec A \times \vec B \|$ by the angle $\arccos{(\| \vec A \| \cdot \| \vec B \|)}$....
HLorenzi's user avatar
  • 175
3 votes
3 answers
4k views

half sine and half cosine quaternions

Something is a little bit unclear to me. In the image below you see that you need to divide the angle by a half. Acccording to wikipedia they say that this is so that I could rotate clockwise or ...
Jordi Ozir's user avatar
1 vote
2 answers
1k 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) &...
Vazrael's user avatar
  • 2,281
0 votes
3 answers
3k views

Rotating a conic section to eliminate the $xy$ term

Problem: Given the equation $$5x^2 + 5y^2 - 6xy - 8 = 0$$ defining a non-degenerate conic section, find a rotation of the variables, such that the cross term $-6xy$ disappears in the new coordinates $...
Alec's user avatar
  • 4,104
0 votes
1 answer
354 views

$n$-dimensional rotation along a 2D arbitrary plane

Given two vectors in $\mathbb{R}^n$, $v_0$ and $v_1$, which define a plane including the origin a rotation along that plane can be defined from $v_0$ to $v_1$. I know the formula for rotation within ...
sheppa28's user avatar
  • 929
28 votes
3 answers
60k 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
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
8 votes
2 answers
2k views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
hyperknot's user avatar
  • 905
8 votes
2 answers
10k views

Given 3 points of a rigid body in space, how do I find the corresponding orientation (aka rotation or attitude)?

Say, I measure the 3D positions, $\mathbf{p_1(t), p_2(t), p_3(t)} \in \mathbb{R}^3$ of three points in space which are all connected by a rigid body at time $t = t_0$. Then, I make a second ...
user1323995's user avatar
7 votes
1 answer
2k views

Complex numbers and their matrix form.

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by $a\mathbf{...
Irish M Powers's user avatar
6 votes
1 answer
3k views

Complex eigenvalues of a rotation matrix

I am struggling with understanding the meaning of complex eigenvalues of a rotation matrix. 1 is always an eigenvalue - that is clear, since all the vectors on the axis of rotation are not effected ...
Whyka's user avatar
  • 1,983
5 votes
2 answers
4k views

Finding a specific Rotation matrix given a known vector

I have two different reference frames: xyz and x0y0z0. Both share the same origin, but there's a rotation between them. My question is: How can I find the rotation matrix of Eulers angles from xyz to ...
mbaggio's user avatar
  • 51
5 votes
1 answer
16k views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
LaTravavax's user avatar
3 votes
2 answers
2k views

Is there a relationship between Rotors and the Rodrigues' rotation formula

I am trying to understand quaternion in general, and it seems like the path to making sense of how they actually work is to first understand rotors and other techniques related to rotations. By ...
Marc Ourens's user avatar
2 votes
2 answers
3k views

n Dimensional Rotation Matrix

So the rotation matrix for 2D is: \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} and one of three BASIC rotation matrices for 3D is: \begin{bmatrix} ...
notMyName's user avatar
  • 202
1 vote
1 answer
6k views

A proof that an orthogonal matrix with a determinant 1 is a rotation matrix

Reading proof(starting on page 5) for item 1 of "Rotation Matrix Theorem" in this doc i'm stuck at understanding its last step. Matrix A being an orthogonal Matrix, at this step the conclusion that A ...
Pooria's user avatar
  • 477
1 vote
1 answer
317 views

Question regarding basis vectors of root reference frame...

Probably my question is rather silly but then again I would rather ask you than going ahead and doing something even sillier. Right, in an old maths book(or at least what remains of it) I was ...
NellyKol's user avatar
1 vote
3 answers
8k 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 $f(x)=x^2$
Danish Exchangestudent's user avatar
0 votes
0 answers
223 views

Rotation and translation of a conic-section

I tried to solve the following exercises, so I want to ask you if my answers are correct. 1) Given the coordinates system $(O'; X'' Y'')$ asociated to the basis $B=[{b_1=\frac{1}{\sqrt{2}}; \...
AaronTBM's user avatar
  • 351

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