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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Show that $g^{-1} \circ f \circ g$ is a translation, if $g$ is a rigid motion and $f$ is a translation

Question: Let $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a rigid motion, and let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a translation. Show that $g^{-1} \circ f \circ g$ is also a ...
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Rigid transformations

Given that $T(x,y)= (y-2,x+3)$ is a rigid transformation how can i express $T = S \circ U$ where S is a translation and $U$ is an orthogonal transformation. It may be a little simple, but it´s been a ...
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How do I rotate a polynomial algebraically?

How do I algebraically rotate a polynomial 90 degrees CW (clockwise) or CCW (counterclockwise) in the $xy$ plane? For example, rotate $f(x) = x^2$ clockwise $90$ degrees. I understand that a CW ...
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Rotation measurements from one frame of reference to another

Been working on some 3D tracking data . Having trouble understanding the relationship between coordinate systems. Lets say I have 3 coordinate systems with the following points: $C_0$ is the global ...
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55 views

rigid measure preserving transformation

I want to proof Lemma 6.7.2 in C.E. Silva's Book "Invitation to Ergodic Theory" (the proof is left as an excercise). A finite measure-preserving transformation $ T $ is said to be rigid if ...
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Is there a 3D shape that is infinitely rotationally symmetrical in exactly 2 axis?

A sphere is completely rotationally symmetrical in all directions. You can apply any combination of roll, pitch and yaw to it and it would be indistinguishable from the sphere you started with. A ...
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Problem with calculating relative orientation

I am using an IMU which provides absolute orientation of the sensors frame $S$ relative to an earth-fixed frame $N$ in quaternion form, $^S_Nq$. In my experiments, I first obtain an initial ...
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Bracing a polygon without triangles

The following Laman graph braces a square without triangles. Stated another way, this is a unit-distance rigid graph without 3-cycles. It seems to be the smallest example of a triangle-free braced ...
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Are 4-chromatic 3-connected unit distance graphs always rigid?

At braced heptagon we determined a graph was rigid. That graph is unit-distance, 3-connected, and 4-chromatic. Here are some more graphs, all extremely constrained. I'm pretty sure they are all rigid ...
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Is this braced heptagon a rigid graph?

In Mathematica, GraphData[{"UnitDistance", {21, 2}}] Is this 42-edge graph rigid? It has chromatic number 4. If it was floppy, that might make it an interesting tool in high-chromatic ...
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Pose estimation by aligning non-collinear points

As shown in Fig.1, consider $3$ line segments of equal length each containing $4$ points. Let $j = 1,\dots,3$ and $i = 1,\dots,4$ be the index for the line segment and the points on the line segment ...
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Euler Angles Rotation Sequence

An arbitrary rotation in space can be constructed through the Euler angles. Imagine an $xyz$ axes, rotate around $z$ by $\phi$ so that the new axes is $\xi\eta\zeta$. Next rotate around $\xi$ by $\...
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Align axis system/transformation with a user defined reference axis system/transformation

I want to align one transformation T1 (ONLY rotation part) with a reference transformation T2. What I do is calculate the delta transformation between T1 and T2 and then multiply delta(by deliberately ...
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Determine the screw axis of a $3D$ rigid motion given by $F(x)=Ax+c$

Determine the screw axis of a $3D$ rigid motion given by $F(x)=\begin{pmatrix}0&1&0\\1&0&0\\0&0&-1\\\end{pmatrix}x+\begin{pmatrix}1\\2\\3\\\end{pmatrix}$ Characteristic ...
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Given a general homogeneous transform, can it be described by DH parameters?

From wikipedia, the general DH transform will have the following form: Given a general homogeneous transform G, I could test ...
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How can I obtain a rotation axis and angular velocity around that axis from angular velocities around the world coordinate axes?

Say I am given the angular velocity of a rigid 3D object in the form $(\omega_x, \omega_y, \omega_z)^T$, where $\omega_x$ describes the angular velocity around the x-axis, $\omega_y$ around the y-axis ...
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2D Linear Acceleration Local Frame Transformation

Looking for information on how to translate a linear acceleration from one frame of reference to another on a rigid body. I have linear accelerations Ay and ...
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1answer
33 views

Unique Cylinder Poses

In the case of Symmetrical objects like spheres, cylinders and cuboids, several poses can produce the same shape of the object (when looking from a certain frame of reference, like from a camera frame ...
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Rigid registration of multiple 3D bodies

Let's assume I have $N$ 3d bodies and I want to register them with a rigid transformation (Rotation and translation). To simplify the problem, let's assume that we are only looking for a rotation. In ...
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Attempted “proof” of relationship between extrinsic and intrinsic rotation sequences

Here is my attempt at a "proof" that 3 intrinsic rotations in the sequence $x$, $y'$, $z''$ is the same as 3 extrinsic rotations but in the reverse order $z$, $y$, $x$. Suppose we have 3 intrinsic ...
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Find rigid body transform $T_b$, given transforms $T_a, T_c$, where $T_c = T_b^{-1}T_aT_b$ for $T_b$. i.e. find change of basis.

Let $\mathbf{T}_a, \mathbf{T}_b, \mathbf{T}_c \in SE(3)$ be rigid body transforms, and; $$ \begin{equation} \mathbf{T}_c = \mathbf{T}_b^{-1}\mathbf{T}_a\mathbf{T}_b. \label{eq_basis} \end{equation} $...
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converting a unit dual quaternion to a form using sin/cos and plucker coordinates

Many papers using dual quaternions for rigid transformations state and use the fact that apparently any unit dual quaternion $$\mathbf{q_0}+\epsilon \mathbf{q_\epsilon}$$ can be written as $$\cos\frac{...
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Rigid body moving orientation relative to helical trajectory

A rigid sphere moves on a trajectory, made up of points that come out of a mechanical simulation. To a visual inspection, the trajectory is helical. I calculate the axis $\boldsymbol{N}$ by using ...
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Confusion in computing relative velocities between different coordinate frames

I am writing a tree structure and I am trying to compute velocity between different frames on my own. But, I feel like my computation is wrong and was wondering if you guys can help me figure out what'...
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Looking for a simple explanation of Singular Value Decomposition in practice

tl/dr: I'm trying to find the best rotation between two 3d point clouds, and all the answers say "use SVD", but I don't have the math background. However, once I get the concept, hopefully I can use ...
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Extract the rotation support of a rigid transformation matrix

Problem I have a rigid transformation matrix, which consists of a rotation and a translation in $ \mathbb R^3 $. I have trouble determining its rotation axis, in particular the support vector of the ...
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52 views

Is there a conventional order for a geometric transformation's operations?

I'm trying to wrap my head around geometric transformations of 3D points (represented as 4x4 matrices) and understanding how to compose and decompose them. In particular I've encountered, in order ...
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Can the composition of two 3D transformations (rotation+scaling) be decomposed as a rotation+scaling?

Let's say I have two rotations matrices (in 3D) $R_1$ and $R_2$, as well as two scaling (of the form $\text{diag}(s_x,s_y,s_z)$) matrices $S_1$ and $S_2$. Then, I can define two spatial ...
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How should I calculate K in Rodrigues' rotation formula from the second order equation?

from Rodrigues' formula we know that : $$K^{2}\left ( 1 - \cos\varphi \right ) + K \sin{ \varphi} + I = R$$ we also know that $K$ should be $$K =\frac{1}{2 \sin {\left ( TR \right )} }\left ( R -...
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How to decompose the three rotations rotation matrix to rotation around a single axis in space?

we know that the rotation matrix of a 3 rotations around XYZ in order, will be : and we also know that the trace of the matrix is : TR = $$\arccos \left ( -1 + \sum_{i=j} R_{ij} \right )$$ if we ...
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How to find both new translations and rotations in a rotated Coordinate system?

i have 3 translations and 3 rotations known in global coordinate system. I need to know these 6 movements in a new CS, that is rotated and translated from the global coordinate system: in which I ...
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If $G:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is distance-preserving, $\exists p_{0}$ and a linear isometry $F$ such that $G(p)=F(p)+p_{0}.$

I need to show that, if $G:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is distance-preserving, $\exists p_{0}\in\mathbb{R}^{3}$ and a linear isometry $F$ such that $G(p)=F(p)+p_{0}.$ My definition of ...
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1answer
52 views

Calculate accelerations at a given point from accelerations at 3 other points on a rigid body

I've measured accelerations on a vibrating rigid body using 3 accelerometers which measure in 3 directions. So I've 3 vectors with three entries measuring accelerations over time at 3 points (1,2,3): ...
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39 views

Sign independency when rotating an inertia tensor with a rotation matrix

I have a set of points that represent a rigid cylinder for which I calculate its rotation in space (x,y,z) using the formula $ \mathbf{J} = \mathbf{R^T} \mathbf{J_0} \mathbf{R} $ with R being the ...
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60 views

closed form solution for Affine transformation between 2 sets of 3 colinear points with a vector attached to each

I want to estimate the scale, rotation and translation between two sets of 3 colinear 3D points. Each 3D point has a direction information (A unit vector attached to it). In case we don't have this ...
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1answer
34 views

Find rotation that maps 5 points to another 5 points in 3D

Suppose I am given five points $v_1,\dots,v_5 \in \mathbb{R}^3$ in 3 dimensions, and five more points $w_1,\dots,w_5 \in \mathbb{R}^3$. I want to find the rotation that maps $v_1 \mapsto w_1, \dots, ...
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268 views

What is the difference between Lie group sim3 and 3d affine transformation?

I am confused between 3D rigid affine (with scale, without shear) transformation and Lie group's sim(3) matrix. 3D affine should be $[sR|t; 0 1]$ \begin{matrix} s\cdot r_{11} & s\cdot r_{12} &...
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100 views

Find mirror location based on its reflection matrix

I have the following transformation matrix TM which is a reflection matrix: ...
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39 views

Relating two coordinate systems given some rigid-body transforms represented in both?

what I'm really after is image registration. However, I will formulate my problem in purely mathematical terms. I have two disjoint sets of 3D points. The points are located on (or in) one solid body,...
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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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831 views

Given Tait–Bryan angles $ x-y'-z^{''} $ (intrinsic), how can i get Tait–Bryan angles $ z-y'-x^{''} $ (intrinsic)?

I've been reading https://en.wikipedia.org/wiki/Euler_angles for hours and cannot figure out how to solve it. https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Euler_angles_.E2.86....
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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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131 views

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 ...