- #1
Dassinia
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Hello
1. Homework Statement
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
The hyperbola equation
x²/k1+y²/k2=1
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
1. Homework Statement
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
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