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

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Rotate a 3D Vector onto Another 3D Vector

I am trying to transform one triangle onto another triangle in 3D space (Right Triangles). My thought was I align the forward and left vectors, then translate the center of one to the other. ...
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0answers
12 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
2
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3answers
40 views

Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
0
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2answers
27 views

Sense of rotation. How would the rotation matrix look like for this “arbitrary” axis?

My first question is how do you define the sense of rotation about an arbitrary axis? Rotations are usually counterclockwise and when referring to rotation with respect to the $x$,$y$ or $z$ axis ...
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0answers
18 views

Local angle to world angle

I am using a digital gyroscope and I am getting very good results with it, only problem is the local angle does not match the world angle (seen by the world). Red = local X-axle Green = local ...
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1answer
19 views

What happens to the underlying geometry when a lower dimension matrix is embedded in higher dimension?

For example, we can represent a rotation in the xy plane as $$R=\left\{ R(\theta)=\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}, 0 \leq \theta ...
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0answers
7 views

Composition of a rotation and a homothetic transformation of different centers?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$. Furthermore let $h_{\lambda,S}$ be the homothetic transformation of center $S\neq \Omega$ and ratio $\lambda$. What ...
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1answer
15 views

Composition of translation and rotation is a rotation, but what is its center?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$.Furthermore let $t_{\vec{v}}$ be the translation by vector $\vec{v}$. Then $$t_{\vec{v}}\circ ...
0
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1answer
88 views

Rotating a point vs rotating coordinate system

Let's suppose we rotate point $P$ by angle $\alpha$ to obtain point $P'$. How to prove that coordinates of point $P$ are $P'$ in the new coordinate system realized by rotating our coordinate axes by ...
2
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2answers
41 views

What is the equation for this wave?

So it would be hard to describe it, it's better to see it yourself: http://physics.info/waves/surface-wave.html (Angular velocity of rotating points is constant I presume) What is it called? What ...
0
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0answers
15 views

Converting points in a right hand z vertical coordinate system to left hand y vertical [migrated]

I have a series of points in space from Fanuc (robot manufacturer). The points are in a right hand system with positive z up. I need to convert this system to Direct X, which is left handed and has ...
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0answers
28 views

Why does $det(R) = +1$ imply right handed frame?

Let $R$ be a rotational matrix in $SO(3)$ so it satisfies $R^TR = I$ Solvng for $det(R^TR) = (det(R))^2 = 1$ yields two solutions Why does $det(R) = +1$ mean that the frame is a right handed frame? ...
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0answers
19 views

Study the caracteristics of the transformation $f=r\circ t \circ h$.

Let $OABC$ be a square with $(\vec{OA},\vec{OC})=\frac{\pi}{2}$. Let $r$ be the rotation of center $B$ and angle $\alpha=\frac{\pi}{2}$, $t$ the translation of vector $\vec{CA}$, $h$ the homothetic ...
2
votes
1answer
57 views

Can someone direct me to prove the following three claims about a rotation group $SO(3)$ and its Lie Algebra?

In class my prof made three claims about a group and its Lie algebra. I cannot find direct reference to these claims because they are delivered in verbatim (im not even sure if I have them jogged down ...
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0answers
18 views

rotation matrix on the 2-sphere

Let $y_i(s): S^2 \mapsto \mathbb{R}$ with $s = (\theta,\varphi)$ for $i =1,\dots,n$ be a set of orthonormal functions on the 2-sphere (more exactly spherical harmonics) and $\beta: \mathbb{R}^3 ...
0
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1answer
31 views

Simple confusion: How can rotation matrix be associative but not abelian?

On the Wikipedia page it is said that the rotation matrix is associative but it also states that rotation matrices are not abelian. The associative property (I think) implies that we have the ...
1
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1answer
26 views

3D vector precise location calculations

I've asked this question before on stackoverflow but people told me that "It requires some pretty serious trigonometry". I'm just a silly programmer, not that smart that I even know anything about ...
1
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1answer
47 views

Transforming coordinate system vs objects

In computer graphics it's pretty common to assume the camera is always positioned at the origin and oriented in one direction. In case we want to move the camera closer to an object in the world ...
1
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0answers
23 views

Finding eccentricity, directrix, foci of diagonal ellipse by rotating it

A problem I'm working on has the equation for a diagonal ellipse $$5x^2 + 5y^2 - 6xy - 8 = 0$$ which can be rotated 45 degrees to get the vertical ellipse $$\frac{x^2}{1}+\frac{y^2}{2^2} = 1$$ The ...
0
votes
3answers
39 views

Rotating a conic section to eliminate the $xy$ term

Problem: Given the equation $$5x^2 + 5y^2 - 6xy - 8 = 0$$ defining a non-degenerate conic section, find a rotation of the variables, such that the cross term $-6xy$ disappears in the new coordinates ...
0
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1answer
63 views

How to translate to a specific point with rotational transformation.

Basically I have two rectangles. ABCD and EFGH EFGH is rotated around it's centre point (X) ABCD has centre point (W) I also know for the sake of this example that EFGH is rotated counter clockwise ...
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2answers
37 views

About rotating a vector around the unit circle and its new coordinates

Where does $\vec e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ go to? Rotate it by an angle $\theta$. Its new coordinates are $\Bigl(\cos\bigl(\theta + \frac {\pi}{2}\bigr), \sin\bigl(\theta + \frac ...
0
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1answer
40 views

Where does the rotation matrix come from?

I looked through many different books, but none of them explain how they derive the matrix below. They just state it as is and move on. Can you explain, please. $ \begin{bmatrix} R(\vec e_1), R(\vec ...
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2answers
69 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
9
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3answers
203 views

Scaling and rotating a square so that it is inscribed in the original square

I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square. The image below is what I'm ...
0
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1answer
31 views

Determining the points of a rotated square.

I have a square with the following four corner points: (0, 0),(100 0),(100 100),(0, 100). The square is then rotated clockwise ten degrees. What is the formula that will allow me to determine its ...
0
votes
1answer
18 views

Find the volume of the body that is created by these intervals

Find the volume of the body that is created by the rotation of these intervals $$0\leq x \leq {\pi\over 2}$$ $$0\leq y \leq (e^t\times sin(t))$$ I have no idea on where to begin or even know how to ...
1
vote
1answer
57 views

Rotating an object correctly when you can only rotate world axis.

This question may be useful to some people, but it is not posed correctly for my particular situation, please see: Simulating simultaneous rotation of an object about a fixed origin given limited ...
1
vote
1answer
31 views

Spin the bottle: Comparing 2 Euler angles

Angle A is a Euler angle that keeps increasing by increments and Angle B is the stopping point of Angle A (think 'Spin-the-bottle' where Angle A is the current angle of the bottle Angle B is the angle ...
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0answers
12 views

Tait–Bryan angle rotations using quaternions

I am attempting to rotate a point $P$ in $\mathbb R^3$ using the Tait–Bryan angles as the inputs, but the math is done by quaternions. This is what I have done so far: Let the Tait-Bryan angles be ...
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2answers
23 views

Drawing a line at a angle and limiting it's end points

I'm trying to draw lines at a specific angle and the algorithm I'm using is : ...
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1answer
79 views

Geometric algebra: Rotors

I've been (slowly) working my way through a book on geometric algebra and have found one part particularly confusing. I can understand the equation $e_1e_2=\exp(e_1e_2 \pi/2)$ Where the substiution ...
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0answers
40 views

How can we split a single rotation into two along orthogonal axes?

I have the following axis system, where the X-Y plane is horizontal and Z points 'up': I have a horizontal plane that I want to rotate so that the angle between it and the XY plane is theta. I ...
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0answers
24 views

Non-standard 3D rotation of a set of points [duplicate]

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 ...
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1answer
48 views

Geometrical calculation to determine size difference between two rectangles when rotating one

I've asked a programming question on StackOverflow here which should give you a good understanding why I'm trying to do this. I'm asking it here because it's now down entirely to the mathematics of ...
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0answers
42 views

Rotation matrix around one coordinate in N dimensions

Probably a very simple question: Given the standard Cartesian coordinate matrix, $$\begin{pmatrix}1 & \\ & 1 & \\ & & 1\\ & & & 1\\ & & & & ...
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0answers
15 views

Euler angles and composition

Suppose that I have two rotations $A, B \in SO(n)$ with Euler angles: $$A = g^{(n-1)}(\alpha^{n-1}) g^{(n-2)}(\alpha^{n-2}) \dots g^{(1)}(\alpha^{1}),\quad B = g^{(n-1)}(\beta^{n-1}) ...
0
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1answer
33 views

Determine similarity between two sequence of quaternions while allowing a degree of freedom around Z axis

A person holds his phone and rotates it in space in a sequence. I am able to obtain a sequence of quaternions from the phone's motion sensors representing the rotation of the phone from the phone ...
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1answer
21 views

rotate a time series with constraints for start and end

I have a time ordered series of data that I can represent in R like: ...
0
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0answers
19 views

How to determine roll-pitch-yaw parameters from homogeneous transformation matrix

We have a transformation matrix $T = \begin{pmatrix} cos(\theta_1) & sin(\theta_1) & 0 & l_1 cos(\theta_1) \\ sin(\theta_1) & -cos(\theta_1) & 0 & l_1 cos(\theta_1) \\ 0 ...
0
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1answer
27 views

Vector rotation

Given a unit vector and an angle of rotation about the unit vector, how do I quickly compute the rotation matrix. I know there must be a simple formula, but I have been unable to find it.
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0answers
19 views

Apply a rotation to Euler Axis angles

I have a camera orientation in world coordinates expressed in a vector containing cameras axis angles relative to the three world $x$, $y$, and $z$ axes (as example $(0,0, 0)$ would be a camera ...
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0answers
38 views

Slope of image side for 3D rotation

I had a new idea for an experimental 3D assembler (not a rasterizer). The idea requires that I get the slope of the top, bottom, left, or right depending on the $z_n$ axis. My idea works on two ...
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0answers
18 views

Express a 90 degree rotation matrix in terms of a 180 degree rotation matrix? (both anti-clockwise)

A = [-1 0 0 ] [ 0-1 0 ] [ 0 0 1 ] B = [0 -1 0 ] [ 1 0 0 ] [ 0 0 1 ] How can i represent B in terms of A?
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0answers
16 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 ...
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1answer
31 views

Yet another n-Dimensional Rotation question: is there a definition for n-Dimensional rotation which is cosine-distance invariant? (TO CLOSE)

(Moved to Searching for a definition for n-Dimensional rotation which is cosine-distance invariant, flagged it to delete it) I'm wondering if there exists a rotation definition by which the vectors ...
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3answers
54 views

How can I rotate a point 45 degrees counterclockwise around any point?

What is a formula (in terms of $x$ and $y$ coordinates) for rotating one point about another by $45$ degrees counterclockwise? I've tried using: $$x'=x\cos(-45^{\circ})-y\cos(-45^{\circ})$$ ...
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0answers
40 views

Represent $90$ degree clockwise rotation about the $z$-axis as a $3\times 3$ matrix

I honestly can't find anything regarding an issue I have with transformational matrices. I understand that this matrix: $$\begin{pmatrix} \cos 90&-\sin 90&0\\ \sin 90&\cos 90&0\\ ...
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1answer
29 views

Know if a 4x4 matrix is a composition of rotations and translations (quaternions)

I am using quaternions to describe 3D transformations. A position in space is representated by a (x,y,z,1) vector, and a transformation by a 4x4 matrix, following quaternions logics as far as I could ...
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1answer
40 views

3D rotational matrix between two spherical co-ordinate systems.

So I have a classical mechanics problem where I have worked out the azimuthal and altitude angle for a vector, I then want to apply rotational matrices so that the vector is realigned with the z axis ...