# Tagged Questions

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 ...

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### 2nd order derivative of Lie group SO(3)

In P.4 of this technical report there is a equation: \begin{align} \left.\frac{\partial^{2}}{\partial \omega_{x}\partial\omega_{y}}(\mathbf{R}_{0}\exp\{J(\omega)\}) \right|_{\omega=0} & = \...
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### Fractional powers of affine matrices

Take a rubber gasket. Make a slice from the middle to the outside, like the first cut in a pie. Because there was some strain in the rubber, the gasket doesn't close into a ring now, it's more like ...
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### Question about injectivity of exponential mapping between SE(3) and se(3)

If we denote $X, Y \in se(3)$, and they have this relationship $$e^X = e^Y$$ is it safe to assume that $X = Y$ for every element? If it is not, may I know the case when it is not? Intuitively, the ...
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### Rigid Motions uniquely specified by triangles?

I'm currently reading the all-familiar "Concepts of Modern Mathematics" by Ian Stewart. At some point in his chapter on motion geometry, Stewart states: "It is a consequence of the two-...
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### Relating/modeling the sensors on a rigid body

Let us assume there are $N$ sensors permanently fixated on a rigid body each measuring the orientation (call it $q_i$) at their corresponding location (call it $p_i$) with respect to a fixed/well-...
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### Euler-Lagrange equation of motion for tensegrity

I have read this paper “Dynamic equations of motion for a 3-bar tensegrity based mobile robot” (1) and this one “Dynamic Simulation of Six-strut Tensegrity Robot Rolling”. 1) http://digital.csic.es/...
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### On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?

Wikipedia has the answer in the case of the Poincaré disk model. When the point $\mathbf{v}$ is translated to the origin, then the point $\mathbf{x}$ is translated to \frac{(1 + 2\mathbf{v} \cdot \...
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### Are triangles rigid in 4 dimensions?

I have read in a couple of sources that a graph with n vertices is rigid in d dimensions if and only if its rigidity matroid has rank nd - d(d+1)/2. C3 (a triangle graph) has a rigidity matroid of ...
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### affine 3D transformation reconstruction

How can we get the affine 3D matrix in case we have the 3D rotation matrix, the 3D translation vector, the scale factors and the shearing factors? A = SHEARING (4,4) * ScaleMatrix (4,4) * ...
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### How to decompose matrix transformations

Let us assume $A$,$B$ and $C$ are known affine transformation matrices in homogeneous 2D space. If it should happen that $C=A^m B^n$ for some unknown $m,n$, is there a way to detect this short of ...
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### Vanishing gradient of a surface $z=f(x,y)$ by a rigid motion

I'm reading the wikipedia pages on some differential geometry topics, e.g. Gaussian curvature. Let's draw our attention to surfaces $z=f(x,y)$ in 3D Euclidian space. The text states the following: ...
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### Move a polygon to a specified position using only allowed rotations, reflections, and dilations

There is a puzzle in Recreational Math in Khan Academy which is very difficult to solve. This puzzle involves using a restricted amount of transformations. The question does not appear to work in the ...
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### “Averaging” transformation matrices?

I have a question on how best to "average" transformation matrices. Say that I have n number of 4x4 transformation matrices, and I wanted to find a matrix that approximated each one of the n 4x4 ...
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### Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
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### Solving System of Equations using transformation rotation

I've never had to post the same question twice, but my last post is getting filled out with work and I'm going about it a different way so I figured i'd try a whole different question So This is the ...
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### Measure of spread of a set problem

I am looking for a good measure of the spread of a set. For example, a cylinder C is more spread out than a sphere of the same volume. So one guess is using diam(A). But in the above example , I ...
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### Prove $\ell_{ki}\ell_{kj}=\delta_{ij}$

Prove $\ell_{ki}\ell_{kj}=\delta_{ij}$ where $\{\hat{\mathbf{e}}_i\}$ and $\{\hat{\mathbf{e}}_i'\}$ are sets of orthonormal basis vectors for $i\in\{1,2,3\}$, $\ell$'s are the direction cosines such ...
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### Rigid body rotation

The problem I am trying to solve is that I am trying to rotate a rigid body and align it to the X axis in 3D space. I have chosen two points on the body (p1, p2). First I move the coordinates to align ...
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### Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? I think I can just ...
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### What are the coordinates of a point on a rigid body after a rotation in 3D Euclidean space, given the initial coordinates and a center of rotation

Main question Let ($x_p$, $y_p$, $z_p$) be the initial coordinates of a point $P$ on a rigid body in a right-handed 3D Euclidean space. Let ($x_r$, $y_r$, $z_r$) be the coordinates of a center of ...
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### How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space?

I have the equation of two 5th degree polynomials which they don`t intersect with each other .Each curve is made of 100 points and these two curves looks similar but there are small differences .I am ...
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### composition of rotation matrices

We wish to construct a general rotation $\mathbf{R}$ of a coordinate system by composing three elementary rotations $\mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_3$, so that a vector $\mathbf{v}$ is rotated ...
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### What curvature conditions make a surface rigid?

Consider a compact surface $S$, possibly with boundary, embedded in $\mathbb{R}^3$, with the induced Riemannian metric. I believe that if $S$ has constant positive Gaussian curvature (that is, $S$ is ...
Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...