9873

The Four-Vertex Theorem

This shows the result of the four-vertex theorem: A simple closed curve has at least four vertices. You can transform the closed curve by dragging the locator. If the caustic extends beyond the window, you can reduce its size.
Let be a smooth plane curve parametrized by arc length , that is, for all . The number is called the curvature of at . A vertex of is a point where . A vertex corresponds to a cusp of the caustic generated by the curve. The theorem implies that the caustic of a general simple closed curce has at least four cusps (for a caustic, see Caustics on Spline Curves).

SNAPSHOTS

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

DETAILS

Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.
Ian R. Porteous, Geometric Differentiation: For the Intelligence of Curves and Surfaces, Cambridge University Press, 1994.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+