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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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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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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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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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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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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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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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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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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38 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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97 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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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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25 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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1answer
164 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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31 views

what is the result of this Integral with polar coordinates?

I can't understand where am I going wrong with this integral. The final answer should be 4 but I get 2/3. Am I wrong or is the teacher wrong? Given this function: $f(x,y)=x+y$ and this domain D ...
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77 views

Transforming FCC, BCC and HCP lattice types to cubes.

I was wondering if it is possible to transform the FCC, BCC and HCP into SC, or simple cubic lattices while preserving the lengths between the nodes? I would like to transform each into this: ...
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1answer
58 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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Reference for Body-and-bar or Body-and-hinge frameworks

I'd like a comprehensive reference for the mathematical theory behind body-and-bar and body-and-hinge frameworks (brief intro), possibly with an emphasis on the latter. There are a large number of ...
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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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113 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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68 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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35 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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90 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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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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125 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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91 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
132 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
52 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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57 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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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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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 ...
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1answer
197 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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291 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
190 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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125 views

How to define the transform?

$y=f(x)$ is continuous and defined for all $x$ real numbers. Point $A(0,f(0))$ is to be moved to on x axis while $f(x)$ is rigid curve and the rigid curve always passes on point $B (x_1,f(x_1))$ ...
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1answer
199 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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1answer
258 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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478 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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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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377 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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3answers
358 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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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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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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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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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 ...
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Rigid Motions - The product of two rotations around different points is equal to a rotation around a third point or a translation

I'm having some difficulty wrapping my head around rigid motions in a plane. In particular, I'm trying to solve this following problem: In a Euclidean plane, show that the product of two rotations ...
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Finding a Rotation Transformation from two Coordinate Frames in 3-Space

The question I'm trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, ...