Dupin's Indicatrix of a Torus

This Demonstration shows how Dupin's indicatrix changes at a variable point on a torus.

In the tangent plane at a point of a surface , Dupin's indicatrix is given by the equation

,

where the , axes coincide with the principal directions at and , are the principal curvatures of at .

• If is an elliptical point, the indicatrix consists of two ellipses (one real and one imaginary).

• If is a hyperbolic point, the indicatrix consists of two conjugate hyperbolas with asymptotes through the asymptotic directions at .

• If is a parabolic point, the indicatrix degenerates into a pair of parallel lines.