We assume that a(t) is an arclength-parametrized C curve that contained in a plane, (we also...

 

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We assume that a(t) is an arclength-parametrized C curve that contained in a plane, (we also know that a(t) in C means that a has continuous derivatives up to third order.) Here we supposet exist inside the bound [a, b]. Now we can add a new"parallel" curve which has B(t) = a(t) + €N(t), e is some constant and greater than 0 a) please prove that ß is a regular curve whenever € is small enough, (for example whenever 0 < e < €o for some €0 > 0. b) Now please represent the curvature of ß in terms of the curvature of a We assume that a(t) is an arclength-parametrized C curve that contained in a plane, (we also know that a(t) in C means that a has continuous derivatives up to third order.) Here we supposet exist inside the bound [a, b]. Now we can add a new"parallel" curve which has B(t) = a(t) + €N(t), e is some constant and greater than 0 a) please prove that ß is a regular curve whenever € is small enough, (for example whenever 0 < e < €o for some €0 > 0. b) Now please represent the curvature of ß in terms of the curvature of a