Disk Flow Applet
Written by Rick Vaughn
Based on the work of Dr. Joel Hass and Dr. Peter Scott
Instructions
- Choose the grid size.
- The fine grid is 3 pixels wide and is the slowest yet most accurate
approximation to a smooth flow.
- The medium grid is 6 pixels wide.
- The coarse grid is 9 pixels wide. Curves collapse quickly, yet still somewhat smooth.
- The super coarse grid is 15 pixels wide. Curves collapse very quickly.
- Set the "leave a trace" box.
- If you leave a trace (the box is checked), the old curves will be left on the screen as the curve collapses. Color changes will track the progress of the curve. Currently, the color changes every 100 iterations of the disk flow.
- If you choose to not leave a trace, the old curve will be erased before the new curve is drawn after each iteration of the disk flow.
- Use the mouse to draw a curve.
- Pressing and releasing the mouse button sets a point for a polygonal line.
- Dragging the mouse enters a continuous curve.
- Click on the "clear" button to erase the current curve and start over.
- Click the "complete" button to make the curve a closed curve.
- Press the "disk flow" button to start the disk flow.
- Press the "disk flow" button again to start/stop the diskflow.
- Press the "clear" button to erase the screen and start over.
- If you do not see a drawing panel above and only see two horizontal lines, then your browser may not be Java enabled.
Click Here to see a temporary version that implements a one-sided flow (convex-hull flow) Note - may be under construction!
Click here for a version that uses an adjustable rectangular grid. Curves collapse to ellipses instead of circles.
Information about the disk flow
The disk flow applet implements a version of a curve shortening algorithm that
appeared in the paper "Shortening Curves on Surfaces" by Joel Hass and
Peter Scott. This article can be found in Topology Vol. 33, No. 1, pp. 25-43, 1994.
The disk flow algorithm curve shortens a curve on a surface by covering it
with a sequence of convex disks and replacing the portion of the curve inside
of each disk with the unique geodesic segment that preserves the end points.
In this application, the surface is a plane and so the geodesic segments are
all line segments. The sequence of disks used is a grid of squares. The curve
is shortened by replacing the portion of the PL curve inside each square with
a line segment inside that same square that preserves the end points. The new
PL curve therefore has its vertices on the current grid of squares. The grid
is then shifted by a third and the process is repeated.
Thank you for visiting!
Please email me your comments or questions:
vaughn@math.ucdavis.edu