Dupin's Indicatrix of a Torus

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.


Contributed by: Sonja Gorjanc and Desana Štambuk (March 2011)
Open content licensed under CC BY-NC-SA



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.