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[Differential Geometry of Curves] Regular Closed Curve function

  1. Nov 14, 2012 #1
    Let m : [0,L] → ℝ2 be a positively oriented C1 regular Jordan curve parametrized with arc length. Consider the function F : [a,b] x [a,b] → ℝ defined by F(u,v) = (1/2) ||m(u) - m(v)||2

    Define a local diameter of m as the line segment between two points p = m(u) and q = m(v) such that:
    The open line segment (p,q) is included in the interior of m and
    F has a local maximum at (u,v)

    (1) Compute the first derivatives of F, expressing them as functions of (m(v) - m(u)) and the unit tangent Tm

    (2) Prove that if the segment [p,q] is a local diameter then it is normal to m at both points p and q

    (3) Assume that (u,v) is such that the open segment (m(u),m(v)) is interior to m, and that it is normal to the curve at both extremities
    (i) Prove that Tm(u) and Tm(v) are collinear. For the rest of the question admit without proof that Tm(v) = -Tm(u)
    (ii) Prove that the Hessian of second derivatives of F at (u,v) can be put in the form
    H = ( 1- k(u)||m(u)-m(v)|| ...........1
    ...................1............... 1-k(v)||m(u)-m(v)|| )
    2. Relevant equations

    Given above

    3. The attempt at a solution

    I can get part 1, but for part 2 I'm not sure how to prove that a local diameter is normal to p and q. I believe it's something to do with the distance between the tangent lines is should be perpendicular?

    Part 3 is quite beyond me I'm not sure how to tackle that one yet
     
  2. jcsd
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