Tagged Questions

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

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0
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1answer
13 views

Given unit quaternions $q_0,q_1$, find $q$ such that $q_1 = q^* q_0 q$

I rotate an object in space and find two orientation (unit) quaternions. $q_0 = {}^{M_2}_{M_1} q$ is the orientation at the 2nd position relative to the 1st position, measured in frame M. $q_1 = ...
-1
votes
0answers
14 views

Find the exact surface area by rotating curve around x axis

Can someone explain step by step how I would solve this problem, I tried and came up with $1/800(17620\cdot\sqrt{401} + 79 \sinh^{-1}(20))$ plugging this into web assign tells me it is incorrect. ...
1
vote
1answer
23 views

3D rotation around arbitrary axis

I have a 3D rotation matrix, R which is a combination of rotations around x-axis , y-axis and z-axis. I know how to calculate n(the arbitrary axis around which a point rotated about theta angle and ...
0
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1answer
15 views

Progresive diagonalisation of a symmetric matrix

Given a symmetric real matrix $A$ there is an orthogonal transformation that brings $A$ to a block diagonal form, where $a \in \mathbb{R}$: $$ RAR^{T} = \left[ {\begin{array}{cc} a & 0 \\ 0 ...
0
votes
1answer
47 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
0
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0answers
24 views

3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
2
votes
1answer
49 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
0
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1answer
27 views

Quaternions $\Leftrightarrow $ Rotations -Conceptual question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve) $ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y ...
0
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0answers
37 views

Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...
0
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0answers
13 views

Rotation about an axis by matrix multiplication

Suppose I have three axis of rotation vectors $\vec{v_1},\vec{v_2},\vec{v_3}$ and angle of rotation as vectors $\theta_1,\theta_2,\theta_3$. Take a vector $P$ then apply rotation around $\vec{v_1}$ ...
0
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0answers
22 views

Quaternion - An equivalent form

Given Data in the problem I have rotation matrices represented by a quaternion $q(t)$ and we are aware of axis of rotation at each point as $\psi(t)$ and angle of rotation $\theta(t)$. I have a ...
0
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0answers
22 views

Calculate rotation and translation of object from corresponding points. NOT affine transformation

I have measured 4 3D points X and corresponding 4 3D points Xp after rotation and translation of object. From equation ...
3
votes
2answers
36 views

Eigenvalues of 3D rotation matrix

I'm having some trouble calculating the eigenvalues for this rotation matrix, I know that you subtract a $\lambda$ from each diagonal term and take the determinant and solve the equation for $\lambda$ ...
1
vote
1answer
27 views

Construct an $Q$ orthogonal using Givens matrices that for all unitary vectors $x$ and $y$ we have $Q^Tx=y$

Find a method to use given matrices to create an orthogonal matrix $Q \in R^{n \times n}$, that for unitary vectors $x,y$ $\in R^{n}$, $$Q^Tx=y$$ The ideia that i have is: take a sucession of Givens ...
1
vote
1answer
15 views

Range of angle in axis-angle representation of rotations

According to Euler, one can represent any rotation in 3D by an angle in the range $[0,\pi]$ and a unit vector representing the direction of an axis of rotation, some details are here. Other possible ...
0
votes
1answer
18 views

Find the volume of the solid obtained by rotating the region about the y-axis

Find the volume of the solid obtained by rotating the region bounded by $y=5\sin(5x^2)$, $0 \le x \le \sqrt{(\frac\pi5)}$ about the $y$-axis. I get the wrong answer using the cylindrical shell ...
0
votes
0answers
13 views

Identity involving $\theta=\arctan(A)$ and $\theta=\arctan(\frac1A)$

I'm working on rotating two system of equations. (I have other questions involving these rotations if you'd like to see exactly the systems I'm talking about. I just don't know how to link them into ...
1
vote
2answers
58 views

Find the volume formed by rotating the enclosed region

Find the volume formed by rotating the enclosed region $y=4\sqrt{x}$ and $y=x$ about $x=17$ I have tried plugging everything into the formula but I can't seem to get the right answer. How do I solve ...
2
votes
2answers
210 views

How to rotate a vector by 90 degree?

Suppose I am given a vector in 2-D as $AB = {(x_1,y_1) ,(x_2,y_2)}$. Now , what will be the co-ordinates if I rotate the vector about the point $(x_1,y_1)$ both clockwise and counter-clockwise ?here ...
0
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0answers
17 views

Quaternion Integration - And conversion to 3D matrix

I have a rotation matrix let us say $R(t)$ and its quaternion $q(t)$. We know already how to convert a quaternion to rotation matrix. Now if I want find $\int R(t) \ dt \tag1 $ can we do that in ...
1
vote
1answer
38 views

Rotations in higher dimensions vs. Spheres

Where my questions stem from: When we study the rotations in a plane or of some specific higher dimensions, there exists a neat approach to represent all the rotations as a spheres $\mathbb S^i$, for ...
0
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0answers
27 views

Probability for matching hexagons number.

I have some interesting math problem. Let say we have nnumber of hexagons. Each hexagon has random numbers in his corners from ...
0
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1answer
25 views

Quaternion , DCM , Euler Angles and Rotation Matrix Differences and when to use?

Quaternion , DCM[Direction Cosine Matrix] , Euler Angles and Rotation Matrix Differences and when to use ? All of the above components can represent rotation , so when to use each of them , best ...
1
vote
1answer
17 views

volume of solid of rotation: finding r

For a region bounded by: $$y=x+4,\;y=16-x^2;\;around\;y=-5$$ I understand that I will be using the 'washer' method: $$V =\int_a^b\pi r^2h$$ But I'm having a hard time finding $$r^2 \text{ for}= ...
1
vote
1answer
42 views

Solving System of Equations using transformation rotation

I've never had to post the same question twice, but my last post is getting filled out with work and I'm going about it a different way so I figured i'd try a whole different question So This is the ...
1
vote
1answer
64 views

Reaching a point B in Cartesian coordinate via Euler angles knows its initial point A Euler angles and Cartesian coordinates

I have a point A:- Known it's Cartesian coordinates (X,Y,Z) and its Euler angle Aka body rotation (R,P,Y) respectively Roll (rotation around X axis) , Pitch (rotaion around Y axis) and Yaw (rotation ...
1
vote
1answer
36 views

Is $SU(2)\times SU(2)=SU(4)$ true? [closed]

I was wondering if $SU(2)\times SU(2)=SU(4)$ In other terms if $Spin(4)=SU(4)$ Thanks!
0
votes
0answers
15 views

Convertion to Quaternion

Specifications and Data We have a 3D rotation function $R(t)_{3\times 3}$ and function ${K(t)}_{3\times 3}$ a matrix function that gives skew symmetric matrix as out puts. It means it holds the ...
0
votes
1answer
20 views

Recovering a vector from the angles it makes with the coordinate axes

I have the following problem: In order to extract features from human joint based 3D data I only consider the angles of the resulting bones (e.g., vector given by shoulder left joint to elbow left ...
0
votes
0answers
9 views

Rotation of an object's acceleration

I want to track things that can rotate using acceleration. The visual involves overlaying two graphs rotated at different angles. This is needed because the accelerometer also rotates with the ...
0
votes
1answer
39 views

3D plane rotation about a line

In three dimensional space we have a plane and a line. These can be oriented in any way. The plane is rotated about the line by n degrees, meaning that originally the position of the plane is fixed to ...
0
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0answers
15 views

Rotation of curves

It is a pretty simple question, but I can only find specialized answers. How can I rotate any curve by any angle in a graph and find the equation that describes it? Is there a methodology that I have ...
0
votes
1answer
29 views

Turntable Photography problem, Concerning set rotation and intervals.

For a project I have to take pictures of an object on a rotating turntable. Setup is as follows: Camera with separate flashes are in front of a turntable taking pictures of a object on the ...
1
vote
1answer
43 views

Finding the volume of the area between two curves when rotated about the y-axis

I keep going down a rabbit hole when answering this question: Find the volume of the area lying in the first quadrant and bounded by the $y$-axis, the curve $y = x^3$ and the line $y = 3x + 2$, ...
0
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0answers
36 views

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 ...
1
vote
1answer
11 views

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 ...
0
votes
1answer
28 views

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 ...
0
votes
0answers
22 views

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 ...
0
votes
1answer
45 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 ...
1
vote
1answer
54 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 ...
0
votes
1answer
62 views

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$ ...
1
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0answers
32 views

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 ...
0
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0answers
19 views

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} ...
1
vote
1answer
38 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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0answers
29 views

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 ...
0
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0answers
16 views

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? ...
1
vote
1answer
55 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 ...
0
votes
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 ...
0
votes
0answers
34 views

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 ...
1
vote
0answers
25 views

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 ...