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

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16
votes
8answers
23k 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 ...
10
votes
6answers
3k 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
0answers
147 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 ...
6
votes
4answers
3k 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
2answers
76 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) ...
8
votes
2answers
21k 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 ...
4
votes
1answer
7k 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 ...
1
vote
1answer
283 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 ...
5
votes
2answers
691 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
638 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 ...
3
votes
1answer
3k 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
81 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 ...
0
votes
2answers
78 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 ...
0
votes
1answer
80 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 ...
0
votes
1answer
1k 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 ...
23
votes
4answers
2k 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 ...
10
votes
2answers
851 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
4answers
2k 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 ...
4
votes
1answer
364 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 ...
4
votes
1answer
164 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
3answers
730 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 ...
0
votes
0answers
20 views

Non-standard 3D rotation of a set of points

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 ...
6
votes
3answers
223 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
223 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 ...
5
votes
5answers
1k 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 ...
4
votes
2answers
279 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 + ...
4
votes
1answer
87 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
399 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
510 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
87 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
110 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
1k 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
950 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
1answer
209 views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
1
vote
2answers
336 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 ...
0
votes
0answers
15 views

Searching for a definition for n-Dimensional rotation which is cosine-distance invariant

I am wondering if it is possible to define a rotation for an $n$-Dimensional space ($n=2,3,4,5,\dots$). Given any vector $\vec v$, and knowing that it should be rotated to ...
0
votes
1answer
251 views

my plane is not vertical, How to update 3D coordinate of point cloud to lie on a 3D vertical plane

I have a bunch of points lying on a vertical plane. In reality this plane should be exactly vertical. But, when I visualize the point cloud, there is a slight inclination (nearly 2 degrees) ...
0
votes
2answers
4k views

A 4x4 homogeneous matrix for a 90 degree rotation about Y axis?

According to my book Rotations through an angle $\theta$ about the $x$, $y$, and $z$ axes are performed using the following transformation matrices. For example, $$ R_x(\theta) = \left[ ...
8
votes
1answer
190 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
107 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 ...
4
votes
1answer
147 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
73 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 ...
3
votes
2answers
434 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
451 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
259 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
10k 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?
1
vote
0answers
69 views

Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship ...
1
vote
1answer
303 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 ...
1
vote
1answer
28 views

Given an axis of rotation and an angle, work out the rotation angles around x,y,z axis

I want to convert from one 3D rotation convention to another. The first convention has an axis of rotation, $\boldsymbol{r}$ and an angle $\theta_r$ to rotate about this axis. The second convention ...