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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Angular Velocity calculation

I am trying to calculate the time derivative of the quaternion from the following paper: Robotics and Biomimetics (ROBIO) See equation 1 below: ...
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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) ...
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44 views

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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Solving the $3$ DOF rotation in a simplified camera model

I have posted this on the programming StackOverflow and I think maybe more suitable to ask the maths experts here. I'm trying to obtain a $3\times 3$ rotation matrix and a focal length parameter in ...
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72 views

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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13 views

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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157 views

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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37 views

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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85 views

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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2answers
57 views

“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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1answer
273 views

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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13 views

The z-axes of two orthoganal cartesian coordinates frames are aligned, then rotated about their x-axes by an angle. How do I calculate that angle?

I am trying to reverse a series of rotations applied to some Cartesian coordinate systems. Two coordinates systems, C1 and C2, are originally oriented with their z-axes aligned but not their x-axes ...
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51 views

Expression of rotation matrix from two vectors

What is the matrix expression of the rotation matrix in 3D which turns a vector $\vec{a}$ into a vector $\vec{b}$, with both vectors given by their coordinates? ($\vec{a} = (a_x, a_y, a_z)$ and ...
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118 views

Inverse of a rigid transformation

I would be grateful for any help with the steps required to complete this calculation. You may assume that I have some experience with matrices from before, but I am obviously no master! So we have ...
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2answers
62 views

Identifying compositions of reflections, and rotations in a hexagon

Let $ABCDEF$ be a regular hexagon that is oriented clockwise (so that a rotation from $A$ to $B$ to $C$ to $D$ to $E$ to $F$ is clockwise). i) Identify $R_{D,120} \circ R_{A,60}$ which are two ...
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2answers
62 views

How to translate a vector and then rotate by a point

I am trying to do this problem: Identify the combination formed by first translating by the vector $(2,0)$ and then rotating by $90$ degrees about $(0,0)$. but I'm a bit confused so, I ...
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1answer
37 views

Let $A,B,C$ be the vertices of a triangle. Find the center of the rotations. $R_{C,\pi} \circ R_{A,\frac{\pi}{2}}$

Let $A,B,C$ be the vertices of a triangle. Find the center of the rotations. $R_{C,\pi} \circ R_{A,\frac{\pi}{2}}$ Hey everyone, I am learning about euclidean rigid motions and my book only ...
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100 views

Second derivatives of rotations

Given an exponential parameterization of a 3D rigid rotation $R\in SO(3)$ by the vector $v = (v_x, v_y, v_z)^T$ I would like to find its second derivatives at the point $v=(0,0,0)$. Using the ...
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is there any similarity transformation property on translation?

Given matrix $A,B,C \in SO(3)$ and the relationship $$A = B^{-1}CB$$ The amount of angle in rotation matrix $A$ are equal to that of rotation matrix $C$, yet not necessarily same principal axis, ...
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32 views

Error metric from affine transformation

I have an affine transformation matrix consisting of a translation and a rotation of a 3D object. I'm developing an algorithm where such a translation should ideally converge to identity, i.e. any ...
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1answer
34 views

Is there a name for this operation of collecting the translation of a point in several directions?

I am writing some code that deals with a 2-dimensional grid and have an operation I am calling project which takes a point and translates it by the grid unit in ...
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1answer
38 views

Determining transformation matrix from six points

Given that I have the locations of three points: p1 = [1.0,1.0,1.0] p2 = [1.0,2.0,1.0] p3 = [1.0,1.0,2.0] ...and I know their transformed counterparts: ...
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Is a rigid cycle a chordal graph?

There are two relevant questions: (1) We know an edge set $C$ is a rigid cycle in $\mathcal{G}_2(n)$ if and only if $|E(C)|=2|V(C)|-2$ and $|F|\leq 2|V(F)|-3$ for every proper subset $F$ of $E(C)$. ...
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Solving $CT = PC$ for transforms in $SE(3)$

I have three transforms: $C$, $T$, and $P$. Each of these transforms consists of 3D rotations and translations. I know $T$ and $P$, and I would like to solve for $C$. They are related by $T = C^{-1} P ...
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84 views

Determine position and orientation of a rigid object, given certain limited informations

I have a rigid 3d object with an unknown position and orientation. I want to determine this pose of the object. On the surface of the rigid object are 4 reference points. I know the spatial ...
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1answer
119 views

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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25 views

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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2answers
53 views

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 ...
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67 views

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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297 views

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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160 views

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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233 views

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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73 views

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 ...
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1answer
577 views

Jacobian of exponential mapping in SO3/SE3

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 ...
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1answer
35 views

Derivative of rigid motion like reflection?

Is it possible to define a derivative for rigid transformations eg. reflection and translation? I am especially interested on reflections shortly $\sigma$. Because I am trying to relate ...
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946 views

Calculate rotation/translation matrix to match measurement points to nominal points

I have two matrices, one containing 3D coordinates that are nominal positions per a CAD model and the other containing 3D coordinates of actual measured positions using a CMM. Every nominal point has ...
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1answer
105 views

Which of these rotation matrices represents a positive rotation in three-space about the y-axis?

This is what Wikipedia says: \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} This is what I think it should ...
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1answer
35 views

Is group of rigid body motion compact?

I believe that group of rigid body motion is not compact. I mean all transformations in $R^3$ that preserve distance. But I need to know how to proof it? From where I should start to prove it?
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A question of H.G. Wells' mathematics

H.G Wells' short story The Plattner Story is about a man who somehow ends up being "inverted" from left to right. So his heart has moved from left to right, his brain, and any other asymmetries ...
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1answer
244 views

To Prove That a Certain Set is a Manifold

Definitions and Notation: Let us write $\underbrace{\mathbb R^n\times \cdots\times\mathbb R^n}_{m \text{ times}}$ as $(\mathbb R^n)^m$. A rigid motion in $\mathbb R^n$ is a function $L:\mathbb ...
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3answers
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What is the precise definition of a rigid shape?

Wikipedia's section on rigid shapes does not appear to actually define what a rigid shape is. Rather it defines 'same shape' and 'rigid transformations' without giving any definitions of what is ...
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1answer
77 views

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 ...
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1answer
38 views

Calculating transformation from origin to point

I have an icosahedron of radius $x$ with 12 vertices at known coordinates. If I have a point at $(0,0,x)$ where $x > 0$ and a vertex of this icosahedron at $(a,b,c)$ how can I find the rotation ...
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113 views

3D Cartesian Transformation

I have a tetrahedron in a 3D Cartesian space. It has two orientations. I know the same three vertices positions (xyz) in the first orientation and the second orientation. I know the position of the ...
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2answers
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Subgroup of motions without a translation?

Let $G \subset M$ be the smallest subgroup of rigid transformations in $\mathbb R^2$ containing a rotation of $1$ radian about $(0,0)$ (call this element $m_1$) and a rotation of $\frac{\pi}{4}$ ...
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1answer
248 views

Estimate for a rigid transform given a set of noisy measurements

I have a set of rigid transforms $\in \mathbb{R}^{4x4}$, where each transform is an approximation to some unknown, "correct" transform. I'm looking for an algorithm to estimate the correct transform ...
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102 views

Solving Generalized Eigenvalue Problem perturbatively

Let me formulate the problem to convey my notation. I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation $$ U_A A\,\,U_A^{-1} = A_{diag}$$ Now the matrix is changed, ...
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1answer
213 views

Finding the rotation transform between coordinate frames in 3-Space given 1 point

I would like to find the rotation transform between two 3D Cartesian coordinate frames knowing only the magnitude and direction of a single vector shown in both frames. The vector passes through the ...
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1answer
62 views

Dimension of nullspace of difference of two rigid transformations

Given distinct proper rigid transformations $A, B \in \operatorname{SE}(n)$, what is the maximum dimension of the nullspace of $A - B$? That is, what is the maximum dimension of $\operatorname{Eq}(A, ...