WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

The Four-Vertex Theorem

reduction

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

α:[a,b]



be a smooth plane curve parametrized by arc length

, that is,

|α(s)|=1

for all

. The number

|α''(s)|=κ(s)

is called the curvature of

. A vertex of

is a point

where

κ'(s)=0

. 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).

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.

I understand and wish to continue anyway »