Dupin indicatrix differential geometry

[Dassinia]

Hello
1. Homework Statement

We define the Dupin indicatrix to be the conic in T_PM defined by the equation II_P(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 II_P(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

Homework Equations

The hyperbola equation
x²/k₁+y²/k₂=1

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

 

[Khashishi]

You are missing some context on what TPM and IIP(v) mean.

 

[Brian T]

Try getting the principle directions by computing the eigenvectors of the shape operator

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