# Invariance of length of curve under Euclidean Motion

• MxwllsPersuasns
In summary, to show that the length of a curve in ℝn is invariant under Euclidean motions, one can start by taking the arc length of the curve and then applying the Euclidean motion. Since the Euclidean transformation preserves distances between points, the arc length remains unchanged. This can be shown by taking the limit of a sequence of functions that depend on distances between pairs of points. Therefore, the length of the curve is invariant under Euclidean motions.
MxwllsPersuasns

## Homework Statement

Show that the length of a curve γ in ℝn is invariant under euclidean motions. I.e., show that L[Aγ] = L[γ] for Ax = Rx + a

## Homework Equations

The length of a curve is given by the arc-length formula: s(t) = ∫γ'(t)dt from t0 to t

## The Attempt at a Solution

I would imagine I would proceed by first taking the arc length of the curve, γ. I'm not sure if I should do this as general and abstractly as possible or if I should give a specific example (i.e., parameterize γ by the parameter (t) perhaps by something like γ(t) = acos(t)i + bsin(t)j -- an ellipse) and then carry out the arc length of that.

My next step would then be applying the euclidean motion to γ, again, if I'm supposed to do this in the abstract then I'm not exactly sure what I should do. If I'm doing a specific example then I would apply A(γ) = R(γ) + a where I believe R is the n-dimensional (in my case n=2) rotation matrix and 'a' is a translation. Now I would multiply the matrix for the curve γ (a 1x2 matrix) with the rotation matrix (a 2x2 matrix) to then get a new curve (Aγ) then I would take the arc length and see that the arc length formula involves γ' and thus the translation constant 'a' wouldn't contribute to the arc length. Then I guess I'd just hope that L(γ) = L(Rγ).

Can anyone tell me if I have the right idea and also whether they think I should do this problem as generally as possible (and then hopefully guide me on the right path on how to do that) or if doing a specific example like γ ∈ ℝ2 would suffice? Thanks so much in advance guys

I don't know what level of detail is expected of you. A Euclidean transformation preserves distances between points. Arc length can be defined as a limit of a sequence of functions, each of which is a function of distances between pairs of points. So after a Euclidean transformation, these distances are not changed. So you taking a limit of the same sequence of functions.

## 1. What is the concept of "Invariance of length of curve under Euclidean Motion"?

The invariance of length of curve under Euclidean Motion refers to the property of a curve or a path to maintain its length regardless of any rigid transformation or motion in a Euclidean space.

## 2. Why is the concept of "Invariance of length of curve under Euclidean Motion" important in science?

This concept is important because it allows us to measure and compare distances and lengths in a consistent and reliable manner, regardless of the position or orientation of the objects involved.

## 3. How is the invariance of length of curve under Euclidean Motion demonstrated in mathematics?

In mathematics, the invariance of length of curve under Euclidean Motion is demonstrated through various equations and proofs that show that the length of a curve remains the same even after applying rigid transformations such as translations, rotations, and reflections.

## 4. Can the invariance of length of curve under Euclidean Motion be applied to real-life situations?

Yes, this concept has practical applications in fields such as computer graphics, robotics, and physics. For example, in computer graphics, this concept is used to ensure that objects in a virtual environment appear consistent in size and shape, regardless of their orientation or position.

## 5. Are there any exceptions to the invariance of length of curve under Euclidean Motion?

In some cases, the invariance of length of curve under Euclidean Motion may not hold true. This can happen when dealing with non-Euclidean geometries, where the concept of distance and length may be different from that of Euclidean space. Additionally, in the case of curved or warped spaces, the length of a curve may change under rigid transformations.

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