The rotations tag has no wiki summary.
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Jacobian matrix of the Rodrigues' formula (exponential map)
I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational.
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1answer
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How do you rotate a vector by a unit quaternion?
Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
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3answers
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Freedoms of real orthogonal matrices
I was trying to figure out, how many degrees of freedoms a $n\times n$-orthogonal matrix posses.The easiest way to determine that seems to be the fact that the matrix exponential of an antisymmetric ...
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1answer
142 views
Understanding rotation matrices
How does $ {\sqrt 2 \over 2} = \cos (45 \text{ degrees })$?
Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function ...
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3answers
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Calculate Rotation Matrix to align Vector A to Vector B in 3d?
I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
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1answer
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Uniform distributions on the space of rotations in 3D
I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions.
The Haar measure on $SO(3)$.
The uniform ...
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1answer
71 views
Angular alignment of points on two concentric circles
I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
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3answers
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How to create 2x2 matrix to rotate vector to right side?
I have vector u=(x,y) and i need to create matrix M: M*u=(1,0).
But that matrix has to rotate vector, instead of keep and ...
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1answer
46 views
Optimal rotation to align a circle with external points
I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
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2answers
116 views
How to solve an overdetermined system of point mappings via rotation and translation
I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$.
I'm looking for a ...
3
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1answer
316 views
Can axis/angle notation match all possible orientations of a rotation matrix?
The rotation group is isomorphic to the orthogonal group $SO(3)$. So a rotation matrix can represent all the possible rotation transformations on the euclidean space $R3$ obtainable by the operation ...
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1answer
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Solid body rotation around 2-axes
I am trying to understand how to describe the rotation of a solid body flying in 3D space. From physics forums, I understand that the rotation of any solid object in space, is around 2 axes ...
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1answer
444 views
Confusion on different representations of 2d rotation matrices
When I first learned about 2d rotation matrices I read that you represented your point in your new coordinate system. That is you take the dot product of your vector in its current coordinate system ...
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2answers
149 views
Minimization on the Lie Group SO(3)
Refering to a question previously asked Jacobian matrix of the Rodrigues' formula (exponential map). It was suggested in one of the answer to only calculate simpler Jacobian ...
8
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1answer
134 views
Do rotations about any three non-collinear axes generate $SO(3)$?
Suppose we have three rotations $r_1$, $r_2$ and $r_3$ in $SO(3)$. Each $r_i$ is a rotation about axis $a_i$ by an irrational multiple of $\pi$.
When considered as unit vectors $a_1$, $a_2$, $a_3$, ...
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1answer
761 views
Efficient and Accurate Numerical Implementation of the Inverse Rodrigues Rotation Formula (Rotation Matrix -> Axis-Angle)
I want to implement the Inverse Rodrigues Rotation Formula (also known as Log map from SO(3) to so(3)), in double precision code (MATLAB is fine for the example) preferably as a 3-parameter vector ...
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2answers
448 views
Aren't asteroids contradicting Euler's rotation theorem?
I am totally confused about Euler's rotation theorem.
Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that:
In geometry, ...
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2answers
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Solution to $AX=XB$ for $3\times3$ rotation matrices.
Given the $3\times3$ rotation matrices $A$ and $B$, find the rotation matrix $X$ that satisfies $$AX=XB.$$
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1answer
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How do I get a tangent to a rotated ellipse in a given point?
I have just graduated from a school you would call High School and even though we talked about tangents to ellipses, we never covered rotated ellipses. So, what I am looking for, is a formula for a ...
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1answer
189 views
Do two rotations densely generate $SO(3)$?
Suppose that you are given two rotations $R_1$ and $R_2$ in $SO(3)$ with the following properties:
$R_1$ is a rotation by an irrational multiple of $\pi$ about the z-axis.
$R_2$ is a rotation by a ...
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2answers
446 views
3 Rotations to unit vector (3D)
I've been trying to solve this problem for some time now, but I could really need some help:
I have 3 rotations (one per axis) for an object, and want to create a unit vector telling me in which ...
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3answers
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Rotate a point in circle about an angle
How should I rotate a point (x,y) to location (a,b) on the co-ordinate by any angle?
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1answer
78 views
Rectangle in rotated bounding rectangle
I'm looking to find the width and height of a rectangle without rotation within a rotated bounding rectangle. I have rotation in degrees and the width and height of the bounding rectangle. Basically ...
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5answers
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What are “clockwise” and “counter-clockwise” in matrix rotation?
I'm learning about the math invovled in PCA. For my purposes here, I'm just trying to understand a 90° rotation matrix. I get the concept of a rotation matrix, but when I look on wikipedia, the ...
