Java Applet: Curvature Flow

Collapse Under Curvature

One illustration of level set methods is the collapse of a curve under curvature. A remarkable theorem shows that any curve moving its normal direction with a speed proportional to its curvature most collapse smoothly to a single point and then disappear. For details, see the section on curvature flow.

As illustration, this Java applet examines this effect. Using the mouse, click points on a curve ,and then click the box to connect the points. The curve will then move according to its curvature and both smooth itself out and disappear. It doesn't matter how many twists and bends are in the initial curve; it will still smooth itself out quickly and then disappear.

Instructions:

Using your mouse, click several points above to define the vertices of a polygon. When you are finished entering the points, press the "Draw Curve" button. This will connect the vertices and begin the Level Set Evolution process. Press "Stop" to end the program before completion. To do it again, reload the page.

Some things to watch out for:

• Long skinny arms shrink the fastest.
• Once the drawing becomes circular, it will slow down as it disappears.lots of skinny arms in the drawing, those will
• If the initial curve is not simple (for example, a figure eight), the algorithm will view it as two separate curves, and each will shrink to a circle and disappears.

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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