# Tagged Questions

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### Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
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### Rotate $xyz$ by use of pitch and yaw around origin

I have a project for a game which uses pitch/yaw for the direction of a players head. The pitch ranges from $0$ to $180$ and the yaw is $0$ to $360$. Yaw modifies $X$ and $Z$, pitch modifies the $Y$, ...
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### Find the angle of rotation about a vector caused by application of a rotation matrix

I have a rotation matrix $R$ and a unit vector $\mathbf{v}$. How can I find the angle of rotation about $\mathbf{v}$ caused by the application of $R$?
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### Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
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### Does R' = R*R'*R^(-1) hold?

Given two 3D rotation matrices ($3\times 3$) $R$ and $R'$, does this equivalence hold?: $R' = R*R'*R^{-1}$ My intuition tells me so, but I can't find a formal proof for it. Thanks.
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### proof for euler-rodrigues formula - matrix form

I need for a matrix representation. Exactly I want to know how to get the Euler-Rodrigues formula in a matrix form like here. Thanks!
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### Obtaining rotation matrix from Euler angles if all three rotations happen at once. Does order of multiplication matter?

I'm having a problem getting my head around Euler Angles. Specifically if I wish to obtain a rotation matrix for a system where pitch, roll and yaw have all changed at once by various values... how ...
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### Geometry of the Cayley Transform

I'm trying to understand the geometry of the Cayley transform. Suppose I have a $3 \times 3$ rotation matrix $R$ (i.e an orthogonal matrix with determinant equal to $1$). Let's ignore the corner case ...
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### Describing Cartesian transformations in Cylindrical Polar Coordinates

I have a question about converting functions defined in Cartesian coordinates to a cylindrical polar system. The particular coordinate transformation that I'm reading about is: ...
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### Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
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### Transformation Matrix for rotation around a point that is not the origin

I need to find the matrix that rotates an arbitrary point around $\begin{bmatrix}5 \\6\end{bmatrix}$ by 35* anticlockwise. I figure I need to first move the plane to centre it at the origin, perform ...
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### How do you find angular velocity given a pair of 3x3 rotation matrices?

Let's say I have two 3x3 rotation matrices R1 and R2, each signifying rotation from the global frame to the local frame. I am also given the time difference t between these two matrices. How would I ...
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### Rotation matrix of a frame with respect to the world frame given angular velocity

I am trying to solve a practice problem for my robotics class. Here is the question: A top is rotating around an axis A with $w_1$ rad/sec and the axis itself is rotating around the vertical with ...
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### Reverse rotation back to original coordinates (Euler Angles)

so in the program I'm trying to write (still, it's a mathematical question) I have a set of coordinates and angles (Euler angles) which represent the place and orientation of an object in space, ...
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### Given the degrees to rotate around axis, how do you come up with rotation matrix?

Given angles (in degrees) to rotate around, $x$-, $y$-, $z$-axis how does one come up with the rotation matrix? For example if you have a point $p$ represented by a vector, how do you rotate it by ...
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### Find rotation angle of given image

At first: our aim is to find the total transformation of left house to the right house. What I did it first is translating the house with the center to the origin. I already found out that the ...
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### How does orthonormal basis rotating work?

When you insert an orthonormal set into the column vectors of a matrix, you create a rotation matrix. I can't understand how this works, by simply placing the the vectors in there you have a rotation ...
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### Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all the I learned about orthogonal matrix is that matrices is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my ...
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### Getting $x,y$ position on an image based on given value

This should be simple but my math skills are really bad ... I have an image of 36 images (6 by 6 matrix). These small images are 36 instances of a direction arrow (like from Google maps GPS), each ...
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### Matrix of transform rotation [solved]

Im trying to create matrix which rotates vector. I have $\vec{g}=(g_1,g_2,g_3);\:g_1\in\mathbb{R},g_2\in\mathbb{R},g_3\in\mathbb{R}$ - it represents gravitation. And $\vec{o}=(o_1,o_2,o_3)$ is vector ...
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### Getting rotation matrix from a vector

I have a vector pointing in some direction and I'm trying to find a matrix $M$ that rotates the vector $v_1=(1,0,0)$ to $v_2=(x,y,z)$, i.e., $M v_1 = v_2$. What is $M$ if $v_1$ and $v_2$ are known? ...
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### I've seen “hyperbolic rotation” - from this: generalization to multisection rotation: is this possible?

This question is more in recreational mathematics area By accident I came across the concept of "hyperbolic rotation" where we use a matrix containing $\cosh$ and $\sinh$ instead of the ...
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### Resolving bearings for tilted structure (vector rotations)

I am trying to find a neat solution to resolving bearings for a tilted structure. For example, if I know that a hub is $268.74^{\circ}$ and offset ($X = -3.542$m, $Y = -1.857$ m, $Z = 2.013$m) with ...
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### Pointing a not stabilized camera using imu data and matrix rotation in the euclidean space

this is my scenario: I have a support with a pan/tilt camera and an imu. This support can be moved by changing the pitch, and roll. From the same position of the support, but independent from the ...
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### Converting a 3x3 matrix to euler angles in various rotation orders

So let's say I have a 3x3 orientation matrix set up like: a b c d e f g h i which I have generated from euler angles in a left-handed, Y-vertical system. ...
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### Rotation matrix in arbitrary dimension to align vector

I was sure it's going to be trivial to do, but then got stuck. Problem - given a vector $u\in\mathbb{R}^d$ in $d$ dimensions, find a rotation matrix $M$, such that (Rotation) $M^T=M^{-1}$, ...
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### Rotation of matrices

I am doing rotation of matrices at the moment, I know that if I want to rotate a point, let's say (2,1) 90 degrees clockwise, I have to multiply the matrix [ 2 1 ] * [0 1, -1 0] , but how do I find ...
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### Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by ...
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### Step in Euler's rotation theorem

I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having ...
### Mapping a plane in $\Bbb R^3$ to $\Bbb R^2$
I have three points that represent a rigid body. The rigid body undergoes a planar transformation in $\Bbb R^3$ due to rotation and translation. I am working with angular velocity with nonzero \$\vec ...