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Dupin indicatrix differential geometry

  1. Nov 10, 2015 #1
    Hello
    1. The problem statement, all variables and given/known data

    We define the Dupin indicatrix to be the conic in TPM defined by the equation IIP(v)=1
    If P is a hyperbolic point show:
    a. That he Dupin indicatrix is a hyperbola
    b/ That the asymptotes of the Dupin indicatrix are given by IIP(v)=1
    , i.e., the set of asymptotic directions.
    c/ That the principal directions are the symmetry axes of the Dupin indicatrix
    d/ Using a symmetry argument and the familiarity of Gaussian curvature along D, show that the asymptotic curves cross D perpendicularly

    2. Relevant equations
    The hyperbola equation
    x²/k1+y²/k2=1
    3. The attempt at a solution
    a/ done
    b/ done
    c/ (Ox) and (Oy) are symmetry axes but how can I determine the principal directions ?
    d/ Don't understand what is D here

    Thanks
     
  2. jcsd
  3. Nov 10, 2015 #2
    You are missing some context on what TPM and IIP(v) mean.
     
  4. Nov 11, 2015 #3
    Try getting the principle directions by computing the eigenvectors of the shape operator

    You also have the " = - 1" equation for the hyperbola.
     
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