# Questions tagged [rigid-transformation]

Rigid transformations are mappings from one reference frame to another reference frame in the Euclidean space $\mathbb R^n$. They comprise translation, rotation (and sometimes reflection). More formally, rigid transformation are elements of the matrix Lie group SE($n$), the specially Euclidean group. If reflection is included, these proper rigid transformations are elements of the matrix Lie group E($n$), the Euclidean group.

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### Type of Transformation given the matrices

I have the following matrices. M1 = 0.9045 -0.3847 -0.1840 10.0000 0.2939 0.8750 -0.3847 10.0000 0.3090 0.2939 0.9045 10.0000 0 0 0 1.0000 M2 = <...
1 vote
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### Fit rigid piecewise linear body to point cloud

I have a set of 3D points, each of which can be considered a point lying inside a rigid body of known dimensions. The actual object is straight cylindrical rods arranged as in the image below. The ...
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### Is there an easier way to derive this equation for this mechanics problem?

We're given a semicircle whose total mass $M$ is evenly distributed along its contour and a little particle of mass $m$ is dropped from its top left corner as in the image below all over an horizontal ...
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### Changing the handedness of a homogeneous 4x4 matrix

I have a 4x4 matrix that represents translation rotation and scaling of a 3D object. I have the issue that my matrix is generated by software that uses a left handed coordinate system but I use a ...
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1 vote
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### Rotate a Translation and a Quaternion around the z axis of an arbitrary pose by an angle theta

I need to implement a rotation in a program but it's 15 years I haven't used rigid body motion maths. I use poses that are described by a translation T and quaternion Q. Everything is expressed in the ...
• 111
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### Finding center of rotation by two point clouds

I have given two point clouds in an arbitrary base. How can I find the center of rotation (or the best base), while minimizing the translations in a rigid transformation? In order to find the ...
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### Rigid body moving orientation relative to helical trajectory

A rigid sphere moves on a trajectory, made up of points that come out of a mechanical simulation. To a visual inspection, the trajectory is helical. I calculate the axis $\boldsymbol{N}$ by using ...
• 560
1 vote
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### Looking for a simple explanation of Singular Value Decomposition in practice

tl/dr: I'm trying to find the best rotation between two 3d point clouds, and all the answers say "use SVD", but I don't have the math background. However, once I get the concept, hopefully I can use ...
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### Extract the rotation support of a rigid transformation matrix

Problem I have a rigid transformation matrix, which consists of a rotation and a translation in $\mathbb R^3$. I have trouble determining its rotation axis, in particular the support vector of the ...
1 vote
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### Is there a conventional order for a geometric transformation's operations?

I'm trying to wrap my head around geometric transformations of 3D points (represented as 4x4 matrices) and understanding how to compose and decompose them. In particular I've encountered, in order ...
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### Can the composition of two 3D transformations (rotation+scaling) be decomposed as a rotation+scaling?

Let's say I have two rotations matrices (in 3D) $R_1$ and $R_2$, as well as two scaling (of the form $\text{diag}(s_x,s_y,s_z)$) matrices $S_1$ and $S_2$. Then, I can define two spatial ...
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### How to find both new translations and rotations in a rotated Coordinate system?

i have 3 translations and 3 rotations known in global coordinate system. I need to know these 6 movements in a new CS, that is rotated and translated from the global coordinate system: in which I ...
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### If $G:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is distance-preserving, $\exists p_{0}$ and a linear isometry $F$ such that $G(p)=F(p)+p_{0}.$

I need to show that, if $G:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is distance-preserving, $\exists p_{0}\in\mathbb{R}^{3}$ and a linear isometry $F$ such that $G(p)=F(p)+p_{0}.$ My definition of ...
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I have a set of points that represent a rigid cylinder for which I calculate its rotation in space (x,y,z) using the formula $\mathbf{J} = \mathbf{R^T} \mathbf{J_0} \mathbf{R}$ with R being the ...