# Questions tagged [rotations]

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

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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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, ...
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### "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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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;...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $R^T R = I$). ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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...
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### 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 ...
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### 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 ...
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### 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 ...
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### $45^\circ$ 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 ...
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### 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 ...
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### 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/...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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