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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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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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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31 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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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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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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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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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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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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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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31 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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33 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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70 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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102 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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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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51 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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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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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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405 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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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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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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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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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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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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103 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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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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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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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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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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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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281 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 ...