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

learn more… | top users | synonyms

9
votes
6answers
2k 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. ...
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) ...
11
votes
6answers
15k 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 ...
7
votes
2answers
13k 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 ...
3
votes
1answer
5k views

Understanding rotation matrices

How does $ {\sqrt 2 \over 2} = \cos (45 \text{ degrees })$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function ...
5
votes
2answers
619 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
3answers
432 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 ...
4
votes
2answers
2k views

Rotating one 3-vector to another

I have written an algorithm for solving the following problem: Given two 3-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$. However, I am not sure if the following ...
0
votes
1answer
785 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 ...
4
votes
1answer
127 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 ...
4
votes
1answer
285 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 ...
3
votes
3answers
581 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 ...
6
votes
3answers
187 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
5
votes
1answer
106 views

Whether matrix exponential from skew-symmetric 3x3 matrices to SO(3) is local homeomorphism?

$SO(3)$ denotes 3x3 rotation matrices. This is Lie group, with corresponding Lie algebra being $\mathrm{Skew}_3$, the space of 3x3 skew-symmetric matrices. The link between them is the matrix ...
4
votes
1answer
73 views

Optimal rotation to align a circle with external points

I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
4
votes
2answers
301 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
3
votes
1answer
446 views

Can axis/angle notation match all possible orientations of a rotation matrix?

The rotation group is isomorphic to the orthogonal group $SO(3)$. So a rotation matrix can represent all the possible rotation transformations on the euclidean space $R3$ obtainable by the operation ...
2
votes
1answer
67 views

How can I break down a rotation of known amount around a known axis into two rotations of unknown amounts around known axes?

I have a vector that's been rotated a known amount about a known axis. I would like to break this rotation down into two separate rotations around known, linearly independent axes where the amounts I ...
2
votes
1answer
100 views

Find rotation that maps a point to its target

I have a 3D point that is rotated about the $x$-axis and after that about the $y$-axis. I know the result of this transformation. Is there an analytical way to compute the rotation angles? $$ ...
2
votes
1answer
825 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 ...
2
votes
1answer
777 views

Confusion on different representations of 2d rotation matrices

When I first learned about 2d rotation matrices I read that you represented your point in your new coordinate system. That is you take the dot product of your vector in its current coordinate system ...
1
vote
0answers
94 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 ...
1
vote
1answer
202 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 ...
1
vote
2answers
233 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 ...
8
votes
1answer
173 views

Do rotations about any three non-collinear axes generate $SO(3)$?

Suppose we have three rotations $r_1$, $r_2$ and $r_3$ in $SO(3)$. Each $r_i$ is a rotation about axis $a_i$ by an irrational multiple of $\pi$. When considered as unit vectors $a_1$, $a_2$, $a_3$, ...
5
votes
1answer
95 views

Could someone explain chirality from a group theory point of view?

While answering this question my interest in the rotation/reflection group was piqued. I personally know very basic group theory, not much more than what a group really is. I understand that the ...
5
votes
1answer
1k 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
112 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
3
votes
2answers
340 views

Solution to $AX=XB$ for $3\times3$ rotation matrices.

Given the $3\times3$ rotation matrices $A$ and $B$, find the rotation matrix $X$ that satisfies $$AX=XB.$$
3
votes
1answer
368 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 ...
3
votes
1answer
235 views

Do two rotations densely generate $SO(3)$?

Suppose that you are given two rotations $R_1$ and $R_2$ in $SO(3)$ with the following properties: $R_1$ is a rotation by an irrational multiple of $\pi$ about the z-axis. $R_2$ is a rotation by a ...
3
votes
3answers
7k views

Rotate a point in circle about an angle

How should I rotate a point (x,y) to location (a,b) on the co-ordinate by any angle?
2
votes
2answers
74 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
1
vote
1answer
331 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
2answers
135 views

Rotating between $3$D frames

Given two frames, is it possible to compute any rotation of the form $$R = R_xR_yR_z $$ that would tranform the frame $A$ into the frame $B$? the rotation will be described by Euler angles as I ...
1
vote
2answers
677 views

3 Rotations to unit vector (3D)

I've been trying to solve this problem for some time now, but I could really need some help: I have 3 rotations (one per axis) for an object, and want to create a unit vector telling me in which ...
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
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
1answer
136 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: ...
0
votes
1answer
63 views

Commutative applying rotations around three axis

Rotating an object in a 3 dimensional space by euler angles might be intuitive but comes with some problems. First, the order of applied rotations around the different axis matters. Second, there is ...
0
votes
5answers
6k views

What are “clockwise” and “counter-clockwise” in matrix rotation?

I'm learning about the math invovled in PCA. For my purposes here, I'm just trying to understand a 90° rotation matrix. I get the concept of a rotation matrix, but when I look on wikipedia, the ...