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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Can the location of the 4th vertex of a projected rectangle be calculated?
In another thread, "Can we compute the location of the unseen point" we are trying to calculate the 2D location in an image, of an arbitrary point Q on a rectangle when we know the 2D ...
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Closest rigid-body pose and velocity along a screw motion to a given pose
Given two rigid-body poses given by dual quaternions ${\bf q}_1$ and ${\bf q}_2$, where ${\bf q}_1 = e^{{\bf v} t}$ is some pose along a screw motion. The screw is given by dual vector (Pluecker ...
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Rotate a frame by the given theta about an axis of another frame
I have two frames ($F_1$ and $F_2$) both are represented wrt frame $O$ (origin frame). Therefore, $T_1$ and $T_2$ are known. Also, I know the frame $F_2$ relative to $F_1$, i.e., $T_{rel}$ is known. ...
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How to prove that rigid bodies have fixed areas during their movements?
a rigid body is a pair of: a set $S$ of points with at least two distinct points on space $\mathbb R^3$ and a special continuous position function for each point of $S$ $f:S \times [t_1,t_2] \to \...
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Rigid transformation and similarity transformation in MLS
I am reading the paper Image Deformation Using Moving Least Squares of Schaefer. I attach the link below:
https://people.engr.tamu.edu/schaefer/research/mls.pdf
In Section 2.3 Rigid Deformations, I am ...
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Constructing transformation of an extended spherical movement
I have a frame of reference positioned on a surface of a sphere with the $z$-axis always pointing towards the center of this sphere, which is at distance $d$.
Now given:
spherical angle $\phi$
...
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Quaternion kinematics transition matrix closed form
For rigid body rotation discrete time kinematics equation is as follows
$R_{k+1} = T_r \cdot R_k$
where:
$R$ is a rotation matrix
$T_r = e^{-[a\times]}$ is a transition matrix
$[a\times]$ means a skew ...
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Aligning a rigid body's two axes to screen axes
I am very new to rigid body transformations.
I would like to align a 3D rigid body such that its first axis $o1$ would align with the display's vertical axis $d1$ and its second axis $o2$ with the ...
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How to find the relationship between the quaternion rate and angular velocity?
I'm trying to find a solid proof of the relationship between the quaternion rate and angular velocity, but I keep running into the same problem.
Theorem:
If frame $B$ is rotating in frame $A$ at a ...
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six orthogonal vectors as the displacements resulted from applying rigid transformations (three translations & three rotations) to a set of points
Given N points represented by a $N\times 3$ matrix:
$$ P=
\begin{bmatrix}
P_{11} & P_{12} & P_{13} \\
P_{21} & P_{22} & P_{23} \\
P_{31} & P_{32} & P_{33} \\
\cdots & \...
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Why are there two different (left) Jacobian definitions of SO(3) and what are the differences?
I read about the exponential map of SE(3), in which the left Jacobian of SO(3) is used. I found two different definitions.
Using the a common notation (rotation axis: $\boldsymbol{\omega}$, rotation ...
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Zero degree plus zero degree is how many kelvin? Addition in Affine spaces [closed]
$0^{\circ} = 273.15$ K
If we consider the transformation which represents the degree to kelvin conversion, $T(x)= x +273.15$. Then it is bijective and can be considered as shifting up the line $y = x$ ...
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Why does a wheel need seven spokes to hold it rigid? (an "inverse problem")
In the biography "King of infinite space: Donald Coxeter, the man who saved geometry" by Siobhan Roberts, the following passage describes an aspect of the subject's relationship with ...
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How do I extract the rotation angle about an specific axis from a rotation matrix.
The question sounds similar to Angle of rotation around arbitrary axis from matrix but it is not.
I don't want an angle-axis extraction, from a 3×3 rotation matrix ${\rm R}$. I know how to do that by ...
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What is the cartesian equation of an ellipse whose center is $(3,2)$ , whose rightmost vertex is $(5,7)$ and whose semi minor axis $3$ units long?
Desmos construction corrected
Briefly, my questions deals with the shifting terms that have to be applied in order to move the blue ellipse to the purple one :
https://www.desmos.com/calculator/...
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Angular velocity as a function of axis-angle derivative
Say I have a timeseries attitude of some body given in an inertial frame as axis-angles pairs $\bf{a}$$(t)$, $\phi(t)$ or alternatively just the 3D vector $\boldsymbol{\alpha}$$(t)$ where
$\...
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Exponentiating only the rotational component of SE(3)
I'm studying the SE(3) group for an engineering application and I'm curious about the difference between $SE(3)$ and $SO(3) \times \mathbb{R}^3$.
Practically, I'd like to optimize in the tangent space ...
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Rotation Jacobian from rotation matrix
Consider a system of rigid bodies, possibly constrained by holonomic and nonholonomic constraints and let $q$ be a set of generalized coordinates, uniquely describing the state of the system. Suppose ...
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Can I combine quaternions like this?
I have a rigid body in the shape of a right angle with 3 reference points on it.
For each pose of the rigid body I know the 3d coordinates of each of the 3 reference points.
I am trying to find the ...
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Orientation of rigid body towed to ship
The array sonar is towed behind a surface ship on a long cylindrical cable that can be kilometers long, keeps the array’s sensors away from the ship’s own noise sources, as shown in Fig 2.
Initially, ...
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How to find quaternion or (pitch, roll, yaw) given a mapping for rotated rigid body?
My question is regarding this answer on finding the orientation of a rigid body given 3 points.
Given $x\mapsto QP^{-1}x+(Q_1-QP^{-1}P_1)$ how do you determine the euler angles? I need pitch, roll, ...
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Intrinsic, Extrinsic, Active, and Passive Rotations
Note: all coordinate vectors mentioned below are column vectors.
I am currently studying properties of rotation matrices using Spong and Vidyasagar's textbook "Robotic Modeling and Control." ...
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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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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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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 ...
