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

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Quaternions- Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represnts the position vector as result of rotation with an angular veclocity $\omega(t)$ in quaternion , then you can make the relationship ...
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Inverse rotation transformations

I'm taking the 2-degree gibmle system and position its alignment point in a arbitrary position (denoted by the axes angles phi for the first degree, and theta for the second). How can I reverse the ...
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Multiplication of Rotation Matrices in quaternion

Given Data and specifications NB : * means multiplication Suppose we need to rotate a point $P = \begin{pmatrix} x\\ y\\ z \end{pmatrix}$ with rotation matrix ${Q}_{3\times3}$ then what we do is ...
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Rotations of form $R_z(a)R_y(b)R_x(c)R_z(-a)R_y(-b)R_x(-c)$

Any proper rotation (in three dimensions) can be expressed using the Tait-Bryan (sometimes called improper Euler) angles in the form $$ R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_x(\psi) $$ where ...
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1answer
33 views

Rotation about a point other than the origin

It is challenging for me to see the rotation of image ABCD to get to Aprime,Bprime,Cprime,Dprime. It is easy to see the translation of the prime figure to the double prime, but not so much the ...
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1answer
49 views

Rotate about the x-axis with respect to dy

How would I rotate the region bounded by $y = 4+x^2,\;x=0,\;y=0,\;and\;x=1\;$ along the x-axis in terms of dy. I have already solved this problem in terms of dx see here Here is the ...
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calculus 2: volume of solid of revolution formed by rotation of region

Find the integral for the volume of the solid of revolution formed by rotating the region $R$ bounded by the curves $y = 4+x^2,\;x=0,\;y=0,\;and\;x=1\;$ about the $x$-axis in terms of $dx $ and $ dy$ ...
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Given a corner, draw a square of known size minimizing vertex distance to a point

I've been going a little crazy trying to solve this what I think is a simple problem. Given: A known corner (~90 degrees) as defined by angle ABC. Segment AB is defined by two points on AB with ...
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Circle rotation number invariant under topological semi-conjugacy.

For a circle homeomorphism $f: S^1 \rightarrow S^1$ we can define the the rotation number $$ \rho(f) = \lim_{n \rightarrow \infty} \frac{1}{n}(F^n(x) - x) \mod 1, $$ for a lift $F:\mathbb{R} ...
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1answer
34 views

Converting non-continuous angle to 360

I have a computer program which outputs an unusual angle system. All angles on the left are $0$ (top) to $-180$ degrees (bottom) and all angles on the right are $0$ (top) to $+180$ (bottom) Is ...
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Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
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help me find the gimbal locks

I have this transformation (x, y, z) |-> (x'', y'', z''). How can the gimbal locks be discerned and where are they? ...
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1answer
38 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
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1answer
13 views

How to multiply a vector and matrix when the matrix includes a translation?

What is the proper way to right multiply an $N$ x $N$ matrix $H$ by an $N$ x $1$ vector $\mathbf{v}$, if $H$ includes a translation vector? For example, say $$H=R-\mathbf{tn}^T$$ where $R$ is a ...
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Quaternion Solution of the Rotation Equation

I am trying to make a connection between a 3-d vector ODE with a quaternion ODE and a possible solution in quaternion. In the following, a vector $v$ in $R^3$ is interpreted as the vector part of the ...
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Calculate rotation on a sphere with given coordinates

I have a sphere with a fixed radius. I have a set of points on that sphere, let's say $p_1, p_2$ and $p_3$ and it's $3$D Cartesian coordinates. I rotated each of the points around the center of the ...
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3answers
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45 degree rotation of the line $y=-3x+1$?

Currently working on problems in a textbook for Senior Maths (Year 11 Maths C, named 'Maths Quest - Maths C for Queensland), however I'm currently at a problem where my answer, despite attempting it ...
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1answer
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Rotating a plane defined by a normal and a distance from the origin around an arbitrary point in 3D space

I have a plane defined by its normal and its distance from the origin. I have a rotation matrix and a point in 3D space around which to do the rotation. What formula will allow me to do the rotation? ...
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Cross product uniqueness

I have following relationship between vectors $A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$ Given Data " ' " means derivative ...
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4answers
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Rotating Frame and Angular Velocity

We have an equation $ \frac{dr}{dt}=\Omega \times \bf r \tag 1$ SPECIFICATIONS $\times$ means cross product,$\Omega$ constant angular velocity,${\bf r}$ is the postion vector of an object Given ...
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2answers
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Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
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When do two configurations of points belong to the same Euler Equivalence Class?

When can I say, of two or more configurations of points in a plane, that they belong to the same Euler Equivalence Class? From Euler's rotation theorem, I gather that two configurations of points are ...
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Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
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1answer
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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 ...
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Finding quaternion that transforms to particular basis

I want to find a quaternion $x \in{\mathbb{H}} $ that transforms (rotates) the $ i,j,k $ basis to a particular basis. In equations: $$ x i x^{-1} = a_1 $$ $$ x j x^{-1} = a_2 $$ $$ x k x^{-1} = a_3 ...
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3answers
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Calculate the area of the ellipsoid that rotates around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...
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2answers
24 views

Rotate and translate a line so that it passes through two given points

I have 2 point and a line segment in 2d space. The line only rotates and translates using its mid point. How do I calculate the translation and rotation required for the line to be touching the 2 ...
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39 views

Rigid body rotation

The problem I am trying to solve is that I am trying to rotate a rigid body and align it to the X axis in 3D space. I have chosen two points on the body (p1, p2). First I move the coordinates to align ...
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2answers
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Finding the matrix of a rotation.

I have to find the corresponding matrix for the following rotation: $R_ \frac{π}{4}$ (1,1) My general matrix formula is: $$\begin{pmatrix}1&0&h\\ 0&1&k\\ 0&0&1 ...
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Calculating rotated relative positions on 2D plane

For a game project, I need to calculate positions of items on a 2D plane relative to the camera. Camera can be rotated, and it's coordinates refer to it's center. In the attached images, ...
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Find path between two attitudes subject to body rate constraints

Here's my problem. I have an initial orientation and angular velocity of a body and a final orientation and velocity occurring at a specified time in the future. I have control over how input ...
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1answer
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Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
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How can I rotate a box?

So, to collide two ships together, I currently use: ...
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Convert Euler angles from one group of rotation axes to another

I have Z-X-Z Euler angles which I would like to convert to X-Y-X Euler angles. What would be the formula for that? The exact choice of source and target rotation axis is not important, I just wish to ...
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How to check if an object is on the line of fire?

In my game all objects are facing right direction by default but they can be rotated and one of my object (let's call it "shooter") needs to know if any object is on its line of fire. If so, it ...
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How can I apply a median filter directly to a time-varying rotation matrix?

I need MatLab script which would take a series of rotation matrices (referring to an actual physical object's orientation) and apply median filter to it to eliminate speckle noise from it. The way ...
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1answer
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How can I refer a 3D pose (position + orientation) to a different coordinate system?

I'm working on a robotics project where all poses and marker positions/orientations are stored as a matrix: $$ \mathbf{P} =\begin{bmatrix} \mathbf{R} & \mathbf{t}\\ ...
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2answers
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Derive a quaternion from three axis

My problem originates from some code that I'm writing to parse an obscure file-type in which a geometric entity is defined in it's own 'local space', and a rotation and translation are provided to ...
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2answers
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Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
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Rotating a cube about an axis through opposite vertices

I have a cube made using CSS transforms that I'm trying to animate rotating about an axis going through 2 opposite vertices. What I have: Initial cube: ...
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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 $T_{ij} = \lambda \delta_{ij}$ ...
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How do I compute corners of geodesic rectangle?

I have the center of a rectangle as a latitude and longitude as well as the length of the sides in meters and the orientation in radians. How can I compute the ...
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Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
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Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
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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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1answer
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Rotate $xyz$ by use of pitch and yaw around origin

I have a project for a game which uses pitch/yaw for the direction of a players head. The pitch ranges from $0$ to $180$ and the yaw is $0$ to $360$. Yaw modifies $X$ and $Z$, pitch modifies the $Y$, ...
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1answer
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How do I rotate a rectangle of latitude and longitude?

I have a rectangle with its corners specified in latitude and longitude. I would like to rotate it about it's centre a certain number of degrees. I was using longitude as an x value and latitude as a ...
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1answer
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Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations?

I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which ...
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1answer
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Rotated arc in $\mathbb{R}$ [closed]

We have got the arc $$A = \{(x,y) \in \mathbb{R} \mid x^2 + y ^2 = R ^2, 0 \leq x \leq R, 0 \leq y \leq R\}$$ and $R$ is positive real number. What is the area of ​​rotational figure obtained if ...
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Abstract question about the rotation in 3D [closed]

I've made the following observersations: Zero-dimmensional entities, points, don't really rotate. One-dimmensional entities, lines, rotate about a stationary point. Two-dimmensional entities, ...