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

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1answer
19 views

How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
1
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1answer
13 views

Rotate a vector into a plane spanned by two other vectors

In an application test that I had to do for a job recently, I was asked the following question (I quote): “Given three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. Compute the rotation ...
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2answers
47 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - ...
3
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1answer
34 views

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?
0
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1answer
18 views

Two body problem (rotation around a fixed central point)

Is there a way which isn't physics related, but just using pure maths to find the solution to the following problem: If i have two lines of different lengths at t=0 overlapping each other. They are ...
0
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0answers
6 views

Can gimbal lock be avoided with Euler angles by resetting the axes?

When using Euler angles to describe rotations of an object can we avoid gimbal lock by simply resetting the axes after each rotation? Alternatively when we conduct rotations can be not just ensure ...
0
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0answers
7 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
0
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0answers
8 views

Confusion about coordinate transforms

Lets say I have a camera aligned with the world coordinates system. I rotate it by 180 degrees around the z axis and then by 20 degrees around its new y axis. I have been reading about Euler angles ...
0
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1answer
13 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...
0
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2answers
17 views

Find center of rotation after object rotated by known angle (2D)

I need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle. Previously I would simply find the center of the object, rotate 180 ...
0
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0answers
27 views

Linear transformation with clockwise rotation on z axis

Let $T$ be a linear Transformation from $\mathbb{R}^3$ to itself such that $T$ is $60^{\circ}$ clockwise rotation with fixed z-axis (i.e, rotate the space according to the z-axis) where $\mathbb{R}^3$ ...
0
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1answer
4 views

Find Z Rotation based on X and Y Vector

I have a vector $(x,y) = (x_2 - x_1, y_2 - y_1)$. I have an arrow pointing to 0 degrees. With vector $(x, y)$, how can I find the number of degrees (0 - 360) that will be the direction the arrow ...
2
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0answers
22 views

Rate of convergence of an irrational rotation

Let $e^{i\phi}, e^{i\theta} \in \mathbb{S}^1$. Let $p(x) = x + 2k\pi$, where $k \in \mathbb{Z}$ is chosen so $p(x) \in [0, 2\pi)$. If we assume that $\phi/\pi$ is irrational, then there exists an ...
1
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1answer
75 views

Is it true that a arbitrary 3D rotation can be composed with two rotations constrained to have their axes in the same plane?

I am interested in decomposing an arbitrary rotation in 3D space into the product of two rotations which are constrained to have their axes in the same plane (for instance x-y plane). Statement of ...
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0answers
10 views

Understanding Check: 3D Rotational forces as a vector

Am I correct in understanding that the magnitude of rotational forces in 3 dimensions can be expressed as a vector, from the origin, whose x, y, and z components represent the magnitude of the ...
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0answers
16 views

Rotation in spherical coordinates (w/o Cartesian)

What are the rotation matrices in polar coordinates? Which matrices I should multiply by a unit vector $(1, \theta, \phi)$ to rotate the latter around basic axes to angle $\beta$? Used notation: ...
0
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1answer
23 views

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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0answers
12 views

Unexpected Asymmetry in Tate-Bryan angles extracted from perturbed Quaternion

I’ve checked some references and the following MATLAB code seems to be correct for converting a quaternion to body roll, pitch, and yaw Tait-Bryan angles respectively. ...
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0answers
17 views

How to calculate translation matrix?

I have a point cloud, which consists of three points. First point cloud has points A(xa, ya, za), B(xb, yb, zb) and ...
0
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1answer
19 views

What does rotation northwards and rotation eastwards mean?

The following quote is causing me trouble: " For instance, suppose we start off at ($0^\circ$N, $0^\circ$W), which is just off the Atlantic coast of equatorial Africa, and rotate $90^\circ$ ...
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2answers
24 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
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0answers
21 views

Position Vector - Using Geometry

Given Data and Specifications in the problem $O_i,O_j$ represent two co-ordinate systems with base orthogonal frame vectors $x_j,y_j,z_j$ and another frame $x_i,y_i,z_i$ with respect to that ...
0
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1answer
69 views

Eigenvectors of a Rotation Matrix

The eigenvector of the rotation matrix corresponding to eigenvalue 1 is the axis of rotation. The remaining eigenvalues are complex conjugates of each other and so are the corresponding eigenvectors. ...
0
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1answer
30 views

Calculus: Volume by rotating curve

R is the region in the first quadrant that is bounded on the left by the y-axis, on the right by the curve $x = \tan(y)$, and above by the line $y = \pi/4$; l is the line $x = 1$. I came up with the ...
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0answers
35 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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0answers
10 views

Rotation in configuration space.

Let $R_\psi$ be the rotation in configuration space around a vector $\bf{e}_\psi$ for an angle $\psi$. How is that the space rotation in configuration space have: ...
0
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1answer
18 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 = ...
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0answers
16 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
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1answer
27 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
votes
1answer
21 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
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1answer
49 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 ...
3
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2answers
68 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 ...
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0answers
33 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
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1answer
62 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
35 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 ...
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0answers
41 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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1answer
30 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 $\vec{v_1}$ ...
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0answers
27 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 ...
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0answers
26 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
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2answers
39 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
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1answer
31 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
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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
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1answer
20 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
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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 ...
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2answers
67 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
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2answers
237 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 ...
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0answers
20 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
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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 ...
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0answers
30 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
37 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 ...