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Questions tagged [rigid-transformation]

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 group. If reflection is included, these proper rigid transformations are elements of the matrix Lie group E($n$), the Euclidean group.

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What is the accuracy of SVD in 3d transformations

I have a triangle $x$ with points $x_1,x_2,x_3\in\mathbb{R}^3$ that were measured at one location. The triangle was then transformed to $\bar x$ where the points were measured again as $\bar{x}_1,\bar{...
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Rotation Matrix: Finding points given one point and attitude

I am trying to find points on a rotated rigid body. The body is a ship which has devices onboard and an antenna. The ship has done a survey on device coordinates. So, the exact locations of the ...
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Subgroups of the group of motions on the plane fixing a point is the conjugate of the group of all orthogonal transformations by translations.

How do I prove that subgroups $O'$ of the group of motions on the plane fixing a point $p$ (say) is the conjugate of the group of all orthogonal transformations by translations i.e. $O' = t_{p}\ O\ {...
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Finite subgroups of motoins always fix some point on the plane.

Let $G$ be a finite subgroup of the group of motoins $M$ on the plane. Then there exists a point on the plane which is left fixed by every element of $G$. The proof of this is sketched by our ...
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motion of a rigid cube

A rigid cube is in motion. At the time depicted in the figure the face $ABCD$ is vertical, the velocity of vertex $A$ is vertical down with value $v$, the velocity of vertex $C$ is vertical up with ...
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why preserving norm is equivalent to preserving inner product in rigid body transformation

Define rigid body motion as following transformation $g$ $$g:\mathbb R^3 \to \mathbb R^3$$ such that $$|g(v)|=|v|,\forall v \in \mathbb R^3$$ $$g(u) \times g(v) = g(u\times v)$$ according to ...
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A slight confusion about a particular step in Euclidean Transformations (Computer Vision)

On page 392 of Computer vision by Simon Prince (http://web4.cs.ucl.ac.uk/staff/s.prince/book/book.pdf), equation 15.2 has the following expression for the Euclidean transformation in homogeneous ...
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Prove that there exist exactly two motions which map two line segments of the same length to each other in a Euclidean Frame

Given two line segments $[P,Q]$ and $[P',Q']$ both in 2D Euclidean frames, with both having the same length, i.e. $d(P,Q) = d(P',Q') \gt 0$ , show that there exists exactly two motions such that $T(P) ...
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Rotation transformation between two frames

My goal is to express the transformation between the black frame F1 and the other one F0 (Green, Red, Violet): All what I know ...
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Given a norm, show a map is Affine

So I have this problem: Let $\|v\|_1=|x|+|y|$ be a norm in the plane, if $v=(x,y)$. Now I'm asked to show that, if $T: \mathbb R^2 \rightarrow \mathbb R^2 $ satisfies $\|T(v)-T(u)\|_1=\|v-u\|_1$, ...
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How to understand the convention of Euler angle?

We have 3 DOF for rotation in 3D space. So to describe an arbitrary rotation, we need to describe its 3 DOF. Euler angle does this by dividing a rotation in 3 steps, first rotate along the Z axis of ...
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How to transform points from a plane to a different plane?

I have a plane $\Pi_1$ expressed as $ax+by+cy+d=0$ and a different plane $\Pi_2$ expressed as $ex+fy+gz+h=0$. I am looking for a transformation which rigidly brings points lying on $\Pi_1$ to points ...
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Seeking 3d rotation

I have a 3d rigid body rotation under which the unit vector $(0, 0, 1)$ becomes the unit vector $(n_x, n_y, n_z)$. I need to find what the vector $(w_x, w_y, 0)$ transforms to under that same rotation....
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How much do I have to rotate an object around X, Y, Z in order to make it parallel to a line

I have a cone-shape 3D object like this: I have a line with a normalized direction vector like this: Normalized vector = (-0.702755, -0.514791, -0.491045) I ...
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SVD - near-zero singular value

I have difficulties to handle singular values close to zero. My SVD implementation $A = U \Sigma V {}^{T}$ performs first an Eigen decomposition of the matrix $A {}^{T} A$. That is done with the QR-...
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Convention for the labelling of parameters of the jacobian of rigid body motion

Assuming we have a matrix representing rigid body motion i.e. SE3 matrix as $$ \begin{bmatrix} r11 & r12 & r13 & r14 \\ r21 & r22 & r23 & r34 \\ r31 & r32 &...
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How to prove that every rigid motion $F:\,\mathbb{R}\to \mathbb{R}$ is 1-to-1? [closed]

A function $F:\,\mathbb{R}\to \mathbb{R}$ is a rigid motion if for all $x,y\in\mathbb{R}$ with $x\neq y$, $\vert x-y\vert = \vert F(x)-F(y)\vert$. Using this definition of rigid motion, prove that ...
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Finding the global maximum of sum-of-exponentials

Problem: find $R$ that maximizes the following: $$f(R) = \sum_{i} k_i \exp^{-y_i^T R x_i}\\ k_i \in \mathbb{R},\; y_i, x_i \in \mathbb{R}^2\\ R \in SO(2)\;\; (\text{i.e. 2D rotation matrix (2x2, ...
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Reference request: Rigid body motion

For a long time I have been wondering if there is some reference in which the motion of a rigid motion is systematically studied from a mathematical perspective; i.e., without using the conventional ...
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Curvature and rigid movements

I am getting stuck on a problem that should actually be very easy. Let $\alpha(t)$ be a birregular curve in 3D space, and $M$ a rigid movement. Let's define $\beta = M \circ \alpha $. The question ...
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Element of $\mathfrak{se}(3)$ is a linear/angular velocity pair?

$SE(3)$ represents the special Euclidean group which can be used to describe the position and orientation of a rigid body in 3D space. It can be defined as $$SE(3)=\{(p,R): R\in SO(3), p\in \mathbb{R}^...
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Rotation matrix about any custom axis

I know the rotation matrices about x, y and z axes but what about rotation about any other axis. I mean if I rotate whole coordinate system clockwise about axis (0,0,0) to (1,1,1) through 120 degrees, ...
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Points fixed by the isotropy group of the restricted Poincaré group in Minkowski space

What points does the isotropy group $SO(3)$ fix in the Minkowskian case? For instance, in oriented Euclidean 3-space the isotropy group $SO(3)$ of the Euclidean group $E(3)$ fixes the origin. What is ...
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Prove that every proper rigid motion in space (R^3) that fixes the origin is a rotation about some axis

I tried to prove this theorem but having hard time to start with. What I have gotten hint is that starting with rigid motion f = La(left multiplication by A) where A is in O(3) (set of all orthogonal ...
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1answer
79 views

Proving that the trace of a plane curve is simetric with respect to its normal vector at $s=0$

Let $b >0$ and $\alpha : (-a, a) \rightarrow \mathbb{R^2}$ be a plane curve parametrized by arc length. Suppose that: $$k(s) = k(-s) \quad \forall s \in (-a,a)$$ where $k(s)$ denotes the curvature ...
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Calculating relationship between two coordinate frames with known relationships between each coordinate frame and a third one

Suppose I have three coordinate frames, W, D and C. I will express all translation vectors as $T$ and rotation matrices as $R$, where $T ∈ \Bbb R^3$ and $R ∈ SO(3)$. I know the translation and ...
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Minimizing a transformation describing the configuration of a rigid body with respect to its vertices

I am solving a least squares optimization. I have a 2D rigid polygon, which can be completely described through a rigid transformation $T = \begin{bmatrix} R_\theta &t \\ 0 &1\end{bmatrix}$ ...
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Metric describing difference between two proper-rigid transformations in 3D

I am estimating a transformation $\hat{T} \in SE(3)$ between two 3-dimensional coordinate systems (COS) $C1, C2$. I know the exact (ground truth) transformation between the two COS, namely $T_{GT} \in ...
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862 views

Rigid motion vs Isometry

Does anyone know the difference between rigid motion and isometry? In the real plane these two definitions coincide, but do always coincide?
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relationship between two planes (or their (N,d) vectors)

I'm reading this article in computer vision and I just can't get my head around eqution(6). The scenario is as follows: We have the pose of a camera in world coordinate system as $T_{w,c}$ so that a ...
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124 views

Convert 3D rotation and translation to new coordinate system

I am given a $3\times 3$ rotation matrix and a $1\times 3$ translation matrix defined in a coordinate system where $x$ points right, $y$ points forwards, and $z$ points up. Now I need to describe ...
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Derivative of rigid transformation using exponential map

I have been looking into Lie algebra in order to compute an optimization to figure out the rigid transformation of a set of points. For now, I have calculated the transformation given an axis-angle ...
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187 views

how to rotate point around a line in 3D

I have a body with origin at point A which is represented in homogeneous coordinate of matrix 4x4 and I would like to rotate it around an arbitrary vector in 3d space. I understand how to rotate the ...
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Model of a rigid body in $SE(3)$

$\require{begingroup}\begingroup \newcommand{\bbeta}{\boldsymbol{\beta}} \newcommand{\bomega}{\boldsymbol{\omega}} \newcommand{\bnu}{\boldsymbol{\nu}} \newcommand{\p}{\boldsymbol{p}} \newcommand{\R}{\...
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Minimum number of points to estimate similarity transformation in 3D

I am studying computer vision and learning about different geometric transformations in 2D and 3D. I understand that the 8-point algorithm requires 8 2D point correspondences to recover the 8 DOF of ...
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Expressing Euclidean isometries using complex variables

Studying for a final in my Euclidean geometry class and I'm having difficulties understanding how my book/professor explained Euclidean isometries. He wants us to understand the classification of ...
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How should the order of application of rotation transformation be interpreted (in PowerPoint)?

For a university assignment, I have a question about rotating a picture in PowerPoint. In PowerPoint, a picture can have four transformations. Rotate right (90°), rotate left (90°), flip horizontally ...
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Find optimal orthonormal transformation of functions?

Let $f,g$ be smooth and integrable functions from $\mathbb{R}^d$ to $\mathbb{R}$. Can someone help with mentioning some papers that tell me how to find the optimal rigid transformation $O^*$ such ...
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2D Square matrix have 8 isometric transformations, what is the rule for non isometric transformation?

I do not have formal formation in mathematics so excuse me in advance if my question is easy. Imagine a 2*2 matrix A = |a b| |c d| I was trying to find ...
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Prove that F is a rigid motion if and only if F is a translation or a reflection.

A function $F: \mathbb{R} \rightarrow \mathbb{R}$ is a a) $\textbf{rigid motion}$ if for all $x$, there exists $y\in \mathbb{R}$ with $x$ not equaling $y$ such that $|x-y| = |F(x)-F(y)|$; b) $\...
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1answer
67 views

Differentiate linear transformation

Looking at the answer of this question: Show that the arc length of a curve is invariant under rigid transformation. I don't understand why if $T$ is linear then $$(T\circ \gamma)'(t)=T\circ \...
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366 views

Khan Academy Transformation Puzzle

This problem on Khan Academy has me stumped. It is part of the transformation puzzles exercise part 3. As the questions are randomized, here it is: While solving the previous transformation puzzles, ...
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117 views

Rotation matrix defined by two Euler angles

I need to find the expressions for $\hat{e_{CX}}$, $\hat{e_{CY}}$ and $\hat{e_{CZ}}$ accordingly to that picture. How can I write a proper axis transformation so I can obtain $\hat{X}$, $\hat{Y}$ and $...
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Spline interpolation in SE(3)

Given: a sequence ${A_i}_{i=1}^n$ of elements of the special Euclidean group, $SE(3)$; a sequence of times ${t_i}_{i=1}^n$, where $t_1 \leq t_2 \leq ... \leq t_n$; instantaneous body frame velocity ...
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Find a rotary reflection in 3D space between two given tetrahedra

My problem is the following: Given two congruent tetrahedra (same side lengths but not equally oriented), I have to find the symmetry plane, the axis and the angle of a rotary reflection that goes ...
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Convention when screw has negative displacement and angle

Any rigid body transformation can be expressed as a screw rotation about and translation along an axis. The screw is usually defined by a line (a unit direction and point on the line), a displacement, ...
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Prove that every rotation is equivalent to two successive reflections (in 3D)

Prove that a rotation about any axis by a finite angle is equivalent to successive reflections in two different planes. Here's what I tried: I assumed two reflection planes (passing through the ...
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What transformation maps $y=\frac{1}{4}x-2$ to $y=-3x+6$?

What transformation maps $y=\frac{1}{4}x-2$ to $y=-3x+6$? I have tried many things, rotating around the origin, reflecting about common lines, and nothing seems to work. Any help is appreciated. ...
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Change of variation under rigid coordinate transformation

Suppose I have a functional $$ E=\int F(y_{1,1},..y_{1,n},y_{2,1}\ldots,y_{n,n})d\boldsymbol{x}\,, $$ where $\boldsymbol{y}:\mathbb{R}^{n}\to\mathbb{R}^{n},\,\boldsymbol{y}(\boldsymbol{x})=\left(y_{...
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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} & = \...