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Questions tagged [rotations]

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

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How to calculate a point after rotation given two unit vectors?

I have two unit vectors: before and after rotation. Point (0, 0, 1) is moved to (-0.42, 0.19, 0.88) after rotation. If I had a point of (-0.066, 0.635, -0.184) before rotation, how it would be ...
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How to get an axis of rotation value without the others axis influencing it?

I have written a PID controller that regulate a Roll angle inside a game engine (using euler coordinates), and it works fine when used "upright". But as soon as my object goes inverted or points in ...
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0answers
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Translate point after rotation relative to different origin

I have 200x380 input image and coordinates (63,146) where (0,0) is top-left: I rotate about the centre some amount of degrees and expand the "canvas" to avoid cropping resulting in larger output ...
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Question about definition of rotation on sphere

I just started M.A. and I do not have enough knowledge about aforementioned concepts. I truly want to know everything about this mapping in details. A rotation of the sphere s^2 is a map r= r_(p,α) ...
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How to use rotational matrices to change orientation of a circle?

Let the following circle be defined by the equation $$\vec{r}(t)= \rho \cos(t) \hat{\mathbf{i}} + \rho \sin(t)\hat{\mathbf{j}}$$ Now suppose the circle is rotated and then translated as follows: ...
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1answer
30 views

what makes quaternion representation of rotations unique? [on hold]

So i'm curious about what actually makes quaternions unique. Quaternions are 4 dimensional and multiplication is non commutative. But what part makes the solution unique? Is it because multiplication ...
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22 views

Determining an equation from a graph

Background: I am trying to make a script for a 2d game that calculates how fast the player moves based on the the direction they are trying moving. To do this, it uses variables that represent ...
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1answer
16 views

How to convert equation of a plane into an equivalent coordinate transformation?

Let $\hat{\mathbf{x}}, \hat{\mathbf{y}},\hat{\mathbf{z}}$ be the unit vectors defining my 'world' coordinates. Consider the following: In this example, $$\mathbf{r}_1 = r_{1,x} \hat{\mathbf{x}} + ...
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21 views

center/axis of rotation 2D [closed]

I feel a little lost. I think my problem is quite simple, but unfortunately, my knowledge of math is also simple. I have two rigid bodies that move relative to each other. I know two points of each ...
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Question regarding $SO_3$ and Skew symmetric matrix, show that $Φ(Av) = A · Φ(v) $for every $A ∈ SO_3(R)$ and $ \forall v ∈ R^3$

Write V for the space of 3 × 3 skew-symmetric real matrices (A) Show that for A $\in$ $SO_3(R)$ and $M \in V$ , $AMA^t \in V$. Write $A · M$ for this action. Ans - This is quite straightforward. (...
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1answer
15 views

Angle required to rotate a polygon towards the direction of a vector

I have a problem where I need to rotate a polygon so it has the same direction as the vector $v_1$ (the pointy head face $y$-axis +ve). I tried a solution where I take two vectors one the $y$-axis: ...
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48 views

If $R:S^{1}\rightarrow S^{1}$ is a irrational rotation, $\{R^{n}([x])\}$ is dense in $S^{1}$ for all points. [duplicate]

Let $\alpha$ a irrational number, and $R:S^{1}\rightarrow S^{1}$ the irrational rotation, i.e., $[x]\rightarrow[x+\alpha]$. I need to prove that, for all $[x]\in S^{1}$, the set $\{R^{n}([x])\}$ is ...
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1answer
23 views

Rotation of a non rectangular hyperbola: equation of hyperbola referred to its asymptotes

I'm looking for a way in which I can rotate a non rectangular hyperbola; in particular I'd like to get the equation of a non rectangular hyperbola referred to its asymptotes. To do it I need to ...
2
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1answer
34 views

Downscale quaternion

I have a problem with quaternions used for rotations. I want to take arbitrary unit quaternions and scale them down if they exceed my goal max rotation. So assume I get some unit quaternion $q = [w, ...
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0answers
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Anchoring matrices in linear equation system for uniqueness

In this book "Articulated Motion and Deformable Objects: 7th International Conference" (1) they describe how to get a unique (non-zero) solution for this homogeneous system of linear equations $MR = [....
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1answer
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Rotating a triangle in 3d specified by the normals

In my surface reconstruction algorithm I need to calculate the right direction of the triangles. The coordinates of the original triangle are known. The original (initial) normal, the new desired ...
0
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1answer
23 views

Scalar relationships between the coordinates of two rotated coordinate systems

I'm wondering how come between the coordinate system $\xi \eta$, rotated -45° with respect to $x y$ coordinate system, and the $xy$ coordinate system there are the following relationships: $$x = \...
2
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2answers
29 views

Find orthogonal matrices satisfying constraints

I have come across the following problem, just wondered if there is an analytical way to solve it: Given arbitrary $\mathbf{C} \in \mathbb{R}^{N \times N} $, Find matrices $\mathbb{R}^{N \times N} \...
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0answers
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Realizing every rotation of a tangent space on a sphere as a parallel transport

I am taking a course on elementary differential geometry, in which we use Do Carmo "Differential Geometry of Curves and Surfaces" as our textbook. I have handed in a written assignment solving - well, ...
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2answers
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Invariant Polynomials under Rotations

Consider a real polynomial $P(x, y)$ in two variables. It is called invariant with respect to the rotation by an angle $\alpha$ if $$ P (x \cos(\alpha) − y \sin(\alpha), x \sin(\alpha) + y \cos(\alpha)...
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1answer
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Rotating a 3D vector by a quaternion?

I am looking into performing a rotation of a 3D vector by a quaternion. I understand this entails pre-multiplying the vector $v$ (made into a "pure" quaternion) with $q$ and then post-multiplying the ...
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Find the rotation matrix R that minimizes $\Sigma(x_i^TR{x'}_i)^2$ given 3D vectors $x'_i$ and $x_i$ for i = 1…n

Basically, the question is about finding the rotation matrix that can align one set of 3D vectors as perpendicular as possible to another set of vectors. My gut feeling is that there should be a ...
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0answers
28 views

How to translate from one plane to another?

Good afternoon all, I hope I posted in the right area which i felt closest to linear algebra more than chemistry. I do apologize if its not! With that being said: I am trying to get Molecule X (from a ...
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1answer
51 views

How do I rotate a bitmap image?

I am trying to write an algorithm to rotate a bitmap image of $n$ by $n$ size by an angle $\alpha$. I know that I have to find a rotation matrix, then perform matrix multiplication of the rotation ...
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1answer
14 views

Connection between rotate / translate / scale as operations in 3D space

This feels like a silly question, what am I missing: I'm a 3D artist working on videogames. Sometimes I make 3D artworks like the ones here: http://samuelthomson.org/blog/2012/06/07/topologic-...
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1answer
58 views

3D rotation defining the intersection of 2 planes

Triangles $DAE$, $DCE$ and $DBE$ form a quadrangle $ABCD$, where $\angle BAD$ and $\angle BCD$ are right angles. I have a scenario in which I want to find the angles $\angle ABD$ and $\angle DBE$, ...
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1answer
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Hadamard transforms seen as rotations in higher dimensions

The Hadamard transform – i.e. the multiplication of a vector $x \in \mathbb{R}^{N}$ (with $N = 2^n$) by the Hadamard matrix $H_n$ – yields another vector $k = H_n x $ in $\mathbb{R}^{N}$. ...
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0answers
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Perspective plane projection based on 2 plane

I have two reference plane(same object) with different known height, captured from the same camera. Reference Plane 0, Reference Plane 1. The camera is tilted and the reference plane is flat. I am ...
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1answer
40 views

Where does this formular for rotating a vector in 3D space around another 3D vector comes from?

I found this formular: $\mathbf{R}_{\vec{n}}(\alpha)\vec{x}=\vec{n}(\vec{n}\cdot\vec{x})+\cos(\alpha)(\vec{n}\times\vec{x})\times\vec{n}+\sin(\alpha)(\vec{n}\times\vec{x})$ here: https://de.wikipedia....
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1answer
37 views

One Quaternion two different 4x4 matrix representations and a same result just multiplying unit quaternions.

Working with different papers I found out two ways on writing Quaternions as 4X4 matrices by "two ways" I'm trying to say that the signs on the coefficients of the matrices are a little bit different ...
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1answer
25 views

Coordinates of 3D vector in rotated coordinate system (without using a matrix)

The problem: There is a vector with coordinates X,Y,Z. This vector is in a coordinate sytem that has been rotated by A degrees along the X axis and B degrees along the Y axis. I would like to know ...
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2answers
29 views

Volume of Rotation of a Square

I have a square with vertices of $(0,5), (0,7), (-1, 6) $ and $ (1,6)$. This is to be revolved around the x axis to find the volume of rotation. How do I go about starting this problem? Thanks in ...
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relation between trace and hat operator (skew-symmetric matrices)

To avoid confusion, let me first introduce the notation (although pretty standard) which is required for the question that I want to ask. Let $\mathsf{GL}(3,\mathbb{R})$ be the set of $3\times 3$ real ...
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0answers
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What is the coordinate value after moving counterclockwise by distance $d$ from a coordinate on the ellipse?

Let me define an ellipse function as follows: Assuming $a \ge b$, $$ f(x,y) = \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} = 1,$$ where $(x_0,y_0)$ is the origin of the ellipse, and $a$ and $b$ are ...
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0answers
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How to find the center of an after-rotated rectangle in 2d-space?

I'm trying to write an algorithm that solves the following problem. And although there is a lot of rectangle geometry questions here on math.stackexchange.com, I have not yet found one that answers ...
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0answers
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equation of rotated cylinder

I would like to determine the equation of a cylinder with its axis on y-axis and rotated around the z-axis. The coordinates of a cylinder of radius r centered at the origin are $ x=r \sin(\phi) $ $...
3
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2answers
39 views

Prove that the rotation of sums is equal to the rotation of products

So the question starts off: Prove $$\ e^{t_1+t_2} = e^{t_1}e^{t_2}$$ E(t) is a unique solution to $\dot{E} = E, E(0) = 1$. Let $E_1(t) = E(t_1 + t)$, and E_2(t) = E(t_1)E(t) $$\dot E_1 (t) = \dot ...
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0answers
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rigid body inertia tensor matrix

We now consider a disk of radius R (modelling a wheel) rolling down an inclined plane with inclination α. We assume that it is rolling without sliding. To describe this motion we use the following ...
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0answers
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Equation of rotated surface

I would like to determine the equation of a disk localized on y-z plane rotated around of the z-axis a $\theta$ angle. Could you tell me how to start? The equation of the disk is $$(y-a) ^ 2 + (z-a) ...
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0answers
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Where does $\cos(pi/2)$ and $\sin(pi/2)$ come from in quaternion rotation? Can you provide a simple unit quaternion rotation example?

I have seen many methods of rotation such as $p' = qpq^{-1}$ and $q = \cos (pi/2),v \sin (pi/2)$ , but I became slightly confused by how a unit quaternion is performed around a vector line.
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1answer
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Solve a system of rotation matrices z-x-z == z-x-y

I have the following equation I would like to solve (I'm looking for: r1, s2 and s3): <...
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0answers
22 views

3D Affine Rotation Matrix from Orthogonal Vectors

How does one define an affine rotation matrix in order to rotate a 3D volume to align with a new coordinate system? The current coordinate system is $\mathbf{x}, \mathbf{y}, \mathbf{z}$ and I want to ...
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0answers
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Lie groups exponential map intuition. [duplicate]

I am having a hard time wrapping the exponential map around my head. So if $G$ is a Lie group we have this map : $exp : T_eG \rightarrow G $ My understanding of lie groups is that they represent stuff ...
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3answers
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Modeling a line in 3D space using length and rotations rather than endpoints

I'm using a cad software which can only create cylinders at the origin based on diameter d and height h. For simplicity sake, in ...
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1answer
13 views

Calculating projected coordinates for rotations on a Cartesian plane

There are four points Blue (4,1), Purple (6,3), Red (9,3) and Green (4,5) plotted on a 1 cm Cartesian grid. I want to rotate the diagram about the Blue point anticlockwise (2nd layout) and clockwise (...
2
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1answer
33 views

Rotation matrix with limited rotation speed

I'm working on a flight-simulator type program, and I'm attempting to make elements (e.g. missiles) that lock-onto and track a target. There are plenty of rotation matrix implementations online that ...
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0answers
10 views

Second derivative of rotation field, Derivation of bendings and torsion

I consider a smooth rotationfield $R:[0,\ell]\times (-\varepsilon, \varepsilon) \rightarrow \operatorname{SO}(3)$, $x \in [0,\ell],\, t \in (-\varepsilon, \varepsilon)$. Then the map $R^T \...
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0answers
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Givens Rotation of diagonal element?

Givens rotations are defined to be zero out the element $a_{ji}$ of $A$ by doing $J(i,j,\theta) A$ and J is defined as the matrix $$J(i,j,\theta)_{k,l} = \cases{ \cos\theta &if $k=l=i$\\ \sin\...
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0answers
49 views

Finding quaternion, representing transformation from one vector to another [closed]

Intro. Previously, I've asked a question on converting rgb triple to quaternion. After that question I've managed to get unit quaternions. Since they were unit ...
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0answers
15 views

Euler angle in two given coordinates

I have two coordinates. One is standard and the other is arbitrary. A rotation is represented as ZYX in the standard coordinates. (yes, It is not exactly a Euler angle. I'm sorry about this) and ...