A string of length ℓ is attached to the point γ(0) of a unit-speed plane curve γ(s). Show that when the string is wound onto the curve while being kept taught, its endpoint traces out the curve ι(s) = γ(s)+(ℓ − s)γ˙(s), where 0 0 for all s. Show that the signed curvature of ι is 1/(ℓ − s).
I cant get to why the formula has (ℓ − s) and not ℓ and also i cant find the signed curvature since if i take the second derivative of ι(s) ill get a third degree derivative of γ(s)