# Homework Help: Dupin indicatrix differential geometry

1. Nov 10, 2015

### Dassinia

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. Nov 10, 2015

### Khashishi

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

3. Nov 11, 2015

### Brian T

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

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