Invariance of length of curve under Euclidean Motion

[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

 

[Stephen Tashi]

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.