Intrinsic 3D Curves

The French mathematicians Joseph Alfred Serret and Jean Frédéric Frenet found a way to represent a parametrized curve by intrinsic equations. At each point of the curve (parametrized by arclength ), three mutually perpendicular unit vectors are defined (called a TNB frame). The tangent shows the direction of motion of the point, the normal points toward the direction in which the curve bends, and the binormal is a vector perpendicular to both. Another two quantities are introduced: curvature to measure how quickly the curve is changing its direction, and torsion to measure how quickly the curve is leaving the TN plane. In this Demonstration, the functions (kappa, for curvature) and (tau, for torsion) can be adjusted using the parameters and and can be chosen to build a ribbon-like surface (in fact, a ruled surface) of a selected width and length, in discrete steps applied times. This is possible because the other edge has the same TNB frame, but displaced.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The Serret–Frenet equations that relate the three vectors with the curvature and torsion of a curve are
[1] R. Rucker. "How Flies Fly: Kappatau Space Curves." (Dec 5, 2014) www.cs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm.
[2] Wikipedia. "Frenet–Serret Formulas." (Dec 5, 2014) en.wikipedia.org/wiki/Frenet–Serret_formulas.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+