# 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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### 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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### 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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1 vote
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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 ...
1 vote
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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 ...
• 145
1 vote
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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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1 vote