Java Applet: General Flow

Motion under General Speed Laws

Overview:

A closed curve moving under its curvature, shrinks to a circle and disappears. This Java Applet provides a tool to examine a more general speed law.

Once again, imagine a curve moving perpendicular to itself with speed F. Curvature flow comes from letting F=-K, where K is the curvature. A more general speed law is given by

(1.-Alpha)

(Alpha)

The two extreme cases are

Alpha=0.0: then the curve uniformly expands, and grows bigger and bigger.
Alpha=1.0: then the curve moves under curvature, and shrinks into a circle and disappears.

Instructions:

To explore various values for Alpha:

Input value for Alpha. It must be between zero and one.
Draw the curve in the box below.

Instructions:

Using your mouse, click several points on the white panel to define the vertices of a polygon. When you are finished entering the points, press the "Draw Curve" button. At any time prior to pressing "Draw Curve" you can change the alpha value by enter a new number and pressing return. Once Draw Curve is clicked, the vertices will connect and the Level Set Evolution process will start. Press "Stop" to end the program before completion. To do it again, reload the page.

Some things to watch out for:

If the evolving curve hits the side of the box, it will keep on expanding.
Starting with a perfect circle and the right choice for "Alpha" the competition between expansion and shrinkage will be balanced, and the curve will just sit there.

If You Want To Learn More:

An introductory resource about level set methods and applications:

NEW BOOK ON FAST MARCHING METHODS AND LEVEL SET METHODS:

Sethian, J.A., Fast Marching Methods and Level Set Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press, 1999.

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