Dupin's Indicatrix of a Torus

ϕ

θ

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

In the tangent plane at a point

p

of a surface

M

, Dupin's indicatrix is given by the equation

κ

1

2

x

+

κ

2

y

=±1

,

where the

x

,

y

axes coincide with the principal directions at

p

and

κ

1

,

κ

2

are the principal curvatures of

M

at

p

.

• If

p

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

• If

p

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

p

.

• If

p

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