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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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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31 views

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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40 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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58 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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22 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
47 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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47 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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25 views

Weighted Rigid Body Transformation

Usually if one talks about rigid body transformation between 2 sets of points, it means: Performing rigid body transformation upon 1 set of points so that the least square error between the 2 sets of ...
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92 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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31 views

Matching two configurations by minimizing angles between pairs of points

I want to match two point configurations by rotation. The configurations are given by two $m$ by $n$ matrices $\boldsymbol A$ and $\boldsymbol B$ with each row representing a point in $\mathbb{R}^n$. ...
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55 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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62 views

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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17 views

Using the difference of the Rigid Transformation between two child frames to compute for the other parent frame such that the difference becomes zero

Given: Transformation Frame tree that tells how all frames are related to each other. The top most frame (/map) is the world frame. It contains two child frames namely map1 and map2. I can translate ...
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89 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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65 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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206 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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21 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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37 views

Transformation in polar coordinate system

I have a point $P \in \mathbb{R}^2$. $P$ in a local Cartesian coordinate is given as $(x,y)$ or alternatively the polar representation is given as $(\rho, \theta)$. This local coordinate is located ...
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506 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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66 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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27 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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314 views

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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149 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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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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72 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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37 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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99 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
50 views

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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144 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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93 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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144 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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56 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, ...
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58 views

Combined length of portions of a line segment

Suppose I have a continuous function $f : X \to \mathbb{R}^n$ (where $X \subseteq \mathbb{R}^n$) that is piecewise rigid, i.e. $X$ has a finite partition $\mathcal{P}$ such that for all $P \in ...
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1answer
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How to find transformation matrix, specifically a rotation, between two given 3d vectors? [duplicate]

How to find transformation matrix, specifically a rotation, between two given 3d vectors? I've found something about it but with quaternions. I don't know anything about quaternions. So it would ...
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Given 3 points of a rigid body in space, how do I find the corresponding orientation (aka rotation or attitude)?

Say, I measure the 3D positions, $\mathbf{p_1(t), p_2(t), p_3(t)} \in \mathbb{R}^3$ of three points in space which are all connected by a rigid body at time $t = t_0$. Then, I make a second ...
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688 views

Jacobian of Reprojection Error

I am writing a program to find the transformation between two sets of 3D points extracted from a moving stereo camera. I am using an 'out of the box' Levenberg-Marquardt implementation to find this ...
3
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1answer
228 views

Fitting Shape in Circle for Shape Classification

I need to classify arbitrary 2D shapes. The classification should be invariant to at least affine transform. To achieve this invariance, I decided to "normalize" each shape by fitting it to a unit ...
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336 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 ...
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1answer
226 views

use homography to rotate around x/y axes

I need to construct a homography out of a 3x3 rotation matrix. I am fundamentally misunderstanding some part of how homographies are constructed. I have been assuming that a homography is ...
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1answer
236 views

Applying a transformation matrix to a matrix of larger dimension, like a grey scale image

I have a new job in image processing and just earned my BS in Physics. I know enough to make a lot of mistakes... I have been reviewing my linear algebra a lot lately, especially the affine ...
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276 views

Rigid motion in curvilinear coordinates

I would like someone to clarify this since it has bedazzled me and can't seem to get a grip on it. Consider a 3D real space and Euclidean coordinates ($x_1,x_2,x_3$), with an associated standard basis ...
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502 views

Equiangular polygon inscribed in rectangle

In a drawing application I am writing, I would like to offer the opportunity for a user to draw an equiangular n-sided polygon inscribed in rectangular bounds drawn by their finger (this application ...
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3answers
220 views

Isometries of $\mathbb{R}^3$

So I'm attempting a proof that isometries of $\mathbb{R}^3$ are the product of at most 4 reflections. Preliminarily, I needed to prove that any point in $\mathbb{R}^3$ is uniquely determined by its ...
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421 views

How can I determine a transformation matrix between two 3D datasets?

I'm working on a computer vision problem: I have a moving camera, and a set of objects (reference points, really) that I'm tracking. The objects themselves are rigid-- they do not move relative to ...
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433 views

What are the requirements for a rotation matrix?

Generally speaking, what are the necessary and sufficient properties of a matrix to make it a rotation matrix? Is det(A) = 1 enough?
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3k views

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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194 views

Projection of the area of a bounded plane over other bounded plane

I have two bounded planes $\pi$ and $\rho$ in three dimensional space. Each plane is bounded by a coplanar rectangle. How can I find the orthogonal projection area of $\pi$ over $\rho$? Thanks in ...
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147 views

about parallel translation (isometry of constant displacement is a translation)

Show that for an isometry $T:{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$, if $$ X \mapsto \mathrm{dist}(X,T(X)) \quad (X \in {\mathbb{R}}^n)$$ is a constant map, $T$ is a parallel translation.
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319 views

correct rotation and translation matrices

I wrote a C++ program that can calculate the magnetic field $\bar{B}$ generated by a circular coil that is placed in the origin, for a given point $\bar{P}$ in 3D ...