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I found this problem in Do Carmos "differential geometry of curves and surfaces".

it asks to show that the length of a parallel curve B to A given by:

B=A-rn

where r is a positive constant, and n is the normal vector, and A is a closed convex plane curve, positively oriented.

is given by

len(B)=len(A)+2*pi*r

The obvious start would be to integrate both sides under a standard parametrization or the curves, but why is it true that integrating rn will give 2*pi*n exactly?

any insight is appreciated.

Thanks!