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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Type of Transformation given the matrices

I have the following matrices. M1 = 0.9045 -0.3847 -0.1840 10.0000 0.2939 0.8750 -0.3847 10.0000 0.3090 0.2939 0.9045 10.0000 0 0 0 1.0000 M2 = <...
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Fit rigid piecewise linear body to point cloud

I have a set of 3D points, each of which can be considered a point lying inside a rigid body of known dimensions. The actual object is straight cylindrical rods arranged as in the image below. The ...
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Is there an easier way to derive this equation for this mechanics problem?

We're given a semicircle whose total mass $M$ is evenly distributed along its contour and a little particle of mass $m$ is dropped from its top left corner as in the image below all over an horizontal ...
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Changing the handedness of a homogeneous 4x4 matrix

I have a 4x4 matrix that represents translation rotation and scaling of a 3D object. I have the issue that my matrix is generated by software that uses a left handed coordinate system but I use a ...
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Rotate a Translation and a Quaternion around the z axis of an arbitrary pose by an angle theta

I need to implement a rotation in a program but it's 15 years I haven't used rigid body motion maths. I use poses that are described by a translation T and quaternion Q. Everything is expressed in the ...
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Finding center of rotation by two point clouds

I have given two point clouds in an arbitrary base. How can I find the center of rotation (or the best base), while minimizing the translations in a rigid transformation? In order to find the ...
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Is there a problem with Goldstein, Poole & Safko's description of Euler Angles?

I have a third edition 9th printing of Goldstein, Poole & Safko's "Classical Mechanics" (GPS) On page 152 fig 4.7 describes a three step transformation $(x,y,z) \rightarrow (\xi,\eta,\...
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A question about the standard euclidean group $\mathbb{SE}(3)$

I have a little perplexity about the following fact: If we consider the standard euclidean group $\mathbb{SE}(3)$, an element $g$ can be represented by a matrix $$\mathbb{SE}(n) \ni g=\begin{pmatrix}R ...
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Trouble with Proof of Sine Sum Formula

The proofs that I've seen of the Sine Sum Formula $ \sin(a + b) = \sin(a)\cos(b) + \sin(b)\cos(a) $ are from Khan Academy and Socratic. Both of them begin with geometric constructions like this: My ...
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Reflection in 3 Dimensions

Say you have a set of $N$ points in $\mathbb{R}^3$ with the centroid at origin with fixed distance between the points (assume a rigid body constraint). Assuming the centroid to be fixed at the origin, ...
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How can a translation preserve a norm?

I want to demonstrate that arc length, curvature and torsion of a parametrized curved are invariant under a rigid motion. However, when trying to do so for the arc length I get $$\left\|\frac{d\alpha}{...
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Hinge or universal joint? [closed]

I am reading the book Convex , Polyhedra, 2005, Springer Verlag , written by A. D. Alexandrov. In chapter 10, he introduces the term hinge mechanism consisting of the edges and vertices of a convex ...
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Finding rotation applied on a 3D point so that it lands to a specific pixel

Suppose we have a 3D point $X=(x,y,z)$. Given the projection matrix of a camera: $$ \begin{matrix} f_x & 0 & c_x\\ 0 & f_y & c_y\\ 0 & 0 & 1 \\ \end{matrix} ...
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Deciding isomorphism of two sets of points given the distances between points of the same set

Consider a set of $m$ points in $\mathbb{R}^n$, $2 \le m \le {n \choose 2}$. We do not know the coordinates of the points, but we know the distances of each point from any other point. However, for ...
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Prove that a rigid motion is surjective and injective

Let $V$ be a finite-dimensional inner product space over $R$ and suppose $f: V\rightarrow V$ is a rigid motion. Prove that $f$ is both injective and surjective. So I know that a rigid motion is an ...
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Existence of rigid transformations $T\colon \mathbb{R}^2 \to \mathbb{R}$

I've been trying to prove no transformation that maps $\mathbb{R}^2$ to $\mathbb{R}$ is rigid, but instead I found a possible candidate for it is $T(v)$ = $\vert\vert v \vert\vert$. However, I remain ...
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A rigid motion of the the plane that does not have a fixed point is either translation or glide reflection.

I want to prove the statement : A rigid motion of the plane that does not have a fixed point is either translation or glide reflection (i.e., a translation followed by a reflection). I want a proof ...
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Do projections of convex sets equal (up to an affine transformation) some intersection with a hyperplane?

Let $C$ be a convex subset of $\mathbb{R}^{n}$ and $C'$ its projection into a k-subspace $H\subseteq\mathbb{R}^{n}$ for $k\leq n.$ We can suppose for simplicity $p\colon\mathbb{R}^{n-k}\times\mathbb{R}...
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Expressing linear and angular acceleration of a rigid body in terms of its homogeneous transformation

I am currently reading "A Mathematical Introduction to Robotic Manipulation" by R. Murray, Z. Li, and S. Sastry to learn about kinematics of rigid bodies. Given a homogeneous transformation $...
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Does this transformation appear to be a rigid motion? Explain

Looking at these two examples I believe a an b both would be considered a rigid motion. A rigid motion is a transformation that preserves distance and angle measures. Since the image and preimage ...
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Best approach (software even) for Rigid registration of two 2D Point Sets with known member correspondences

Let's assume that I have in 2D space, an initial set of N points. I take these points, I move them a bit and I consider their new positions as a new Set. Given that I know the point correspondences (i....
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quaternions transformation

I'm working with IMU providing me with their orientation to a common global coordinate system. I have created a segment coordinate system from a static posture with know orientations. I have placed ...
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Rotation matix offset

I need to calculate the relative orientation between two IMU sensors. The Imu sensors provide me with the orientation relative to a global coordinate system (the same). I place the 1 imu on the ...
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prove wallpaper groups are not isomorphic

In the chapter I am reading on wallpaper groups, it outlines proofs that all the wallpaper groups are not isomorphic and hence different. But I do not fully see why what they are saying is true. For ...
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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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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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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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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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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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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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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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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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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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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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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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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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