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ck29205325

Member

Registered: 2009-09-20

Posts: 5

property of curvature

Problem:
Let r:(a,b)->R^2 be a regular parametrized plane curve. Assume that there exists t0,a<t0<b,such that the distance |r(t)| from the origin to the trace of r will be a maximum at t0. Prove that the curvature k of r at t0 satisfies |k(t0)|>=1/|r(t0)|.

It's actually an exercise in the text "Differential Geometry of Curves and Surfaces" by Manfredo Do Carmo.

ck29205325

Member

Registered: 2009-09-20

Posts: 5

Re: property of curvature

Is there any suggestions? plz

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: property of curvature

Think of a circle centered at 0 of radius |r(t0)|, what is the radius of curvature?

Now think of your path, r(t), as it approaches the point r(t0).  Where does it approach from (in the circle?  Outside of the circle? On the boundary?), and given this, what can you conclude about it's radius of curvature?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."