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

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41
votes
13answers
74k 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 the previous normal of the triangle and the current normal of the triangle along with both the current ...
6
votes
3answers
1k 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 ...
11
votes
6answers
6k 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. ...
1
vote
1answer
203 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 ...
1
vote
3answers
1k 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 ...
2
votes
1answer
2k 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 $\arccos{(\| \vec A \| \cdot \| \vec B \|)}$....
26
votes
9answers
2k 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 ...
6
votes
2answers
1k 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, ...
5
votes
5answers
7k 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 ...
5
votes
3answers
2k 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 ...
1
vote
1answer
375 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 ...
0
votes
2answers
166 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) &...
14
votes
3answers
45k 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 ...
11
votes
2answers
15k 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? ...
5
votes
1answer
7k 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 ...
8
votes
2answers
2k views

Combining Two 3D Rotations

Every rotation in 3D space can be defined by a rotation axis and an angle. Now let's say we have two rotations $R_1 (\text{(axis)}_1, \text{(angle)}_1)$, $R_2 (\text{(axis)}_2, \text{(angle)}_2)$. I ...
6
votes
3answers
1k views

Freedoms of real orthogonal matrices

I was trying to figure out, how many degrees of freedoms a $n\times n$-orthogonal matrix posses.The easiest way to determine that seems to be the fact that the matrix exponential of an antisymmetric ...
5
votes
2answers
494 views

How to find the shift that minimizes the difference between two vectors?

I am looking for a efficient way to find the value of k that minimizes $\sum(s_t - b_{t+k})^2$ where $s$ and $b$ are N-dimensional vectors and the values are wrapped around like this: $b_{t+k} := ...
2
votes
1answer
581 views

Closed-form for eigenvectors of rotation matrix

For matrices that are elements of $SO(3)$ is there a formula for the eigenvectors corresponding to the eigenvalue $1$ in terms of the entries of the matrix?
1
vote
1answer
465 views

Rectangle in rotated bounding rectangle

I'm looking to find the width and height of a rectangle without rotation within a rotated bounding rectangle. I have rotation in degrees and the width and height of the bounding rectangle. Basically I'...
1
vote
1answer
946 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
1
vote
0answers
291 views

Rotating an $n$-dimensional hyperplane

Let $\mathcal{H}: \mathbf{x}^T\mathbf{w}+b=0$ be a hyperplane in the $n$-dimensional Euclidean space of column vectors. Is there a way of "rotating" the above hyperplane such that it coincides with ...
0
votes
2answers
185 views

Find the axis of rotation from the rotation matrix.

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ...
5
votes
1answer
2k views

Efficient and Accurate Numerical Implementation of the Inverse Rodrigues Rotation Formula (Rotation Matrix -> Axis-Angle)

I want to implement the Inverse Rodrigues Rotation Formula (also known as Log map from SO(3) to so(3)), in double precision code (MATLAB is fine for the example) preferably as a 3-parameter vector ...
3
votes
1answer
702 views

How do I get a tangent to a rotated ellipse in a given point?

I have just graduated from a school you would call High School and even though we talked about tangents to ellipses, we never covered rotated ellipses. So, what I am looking for, is a formula for a ...
2
votes
4answers
23k 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 ...
1
vote
3answers
67 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 $f(x)=x^2$
0
votes
1answer
2k views

Determine whether or not a point lies within a rotated rectangle

I need some maths help for a 2D game I am programming. In this game I have a rectangle, specified by its centers' X and Y coordinates, and its width and height. I then rotate this rectangle via ...
0
votes
1answer
46 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? ...
68
votes
2answers
804 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 ...
26
votes
4answers
3k 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 ...
5
votes
4answers
4k views

What's the best 3D angular co-ordinate system for working with smartphone apps

This is very much an applied maths question. I'm having trouble with Euler angles in the context of smartphone apps. I've been working with Android, but I would guess that the same problem arises ...
15
votes
1answer
161 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 ...
11
votes
3answers
1k 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 ...
10
votes
3answers
2k 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 ...
5
votes
2answers
329 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism $f:S^...
1
vote
3answers
588 views

Minimization on the Lie Group SO(3)

Refering to a question previously asked Jacobian matrix of the Rodrigues' formula (exponential map). It was suggested in one of the answer to only calculate simpler Jacobian $\left.\frac{\partial}{...
5
votes
1answer
19k 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 ...
4
votes
1answer
566 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
9
votes
7answers
3k 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 ...
5
votes
2answers
1k 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{...
4
votes
2answers
729 views

Interpretation of eigenvectors of cross product

If we fix a non-zero vector $\boldsymbol{v}\in\mathbb{R}^3$, then the linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ has trivial eigenvectors $\boldsymbol{x}_1=t\boldsymbol{v}$ (...
4
votes
1answer
151 views

Mean value of the rotation angle is 126.5°

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
4
votes
1answer
202 views

Angular alignment of points on two concentric circles

I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
3
votes
1answer
2k views

Solid body rotation around 2-axes

I am trying to understand how to describe the rotation of a solid body flying in 3D space. From physics forums, I understand that the rotation of any solid object in space, is around 2 axes ...
3
votes
3answers
1k views

How to create 2x2 matrix to rotate vector to right side?

I have vector u=(x,y) and i need to create matrix M: M*u=(1,0). But that matrix has to rotate vector, instead of keep and ...
3
votes
1answer
5k views

rotating 2D coordinates

I've tried googling this, but I always end up somewhere that just says it's easy. Anyhow, I have a coordinate system, where I need to rotate a bunch of points. It's all 2D. Coordinates varies and so ...
2
votes
2answers
69 views

Find the Vector in the New Position Obtained by Rotation

The vector $\vec{OP}=\hat{i}+2\hat{j}+2\hat{k}$ turns through a right angle,passing through the positive $x-$axis on the way.Find the vector in the new position. Let the new position of the vector ...
0
votes
1answer
2k views

Euler angles to rotation matrix. Rotation direction

So we have a 2D rotation matrix for counterclockwise (positive) angle "$a$": $\begin{pmatrix} \cos(a) & -\sin(a) \\ \sin(a) & \cos(a) \end{pmatrix}$. For clockwise (negative) angle: $\begin{...
0
votes
0answers
28 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...