Figure Captions

1. Geometry of the geodesic curvature vector. 2. Geometry of the constant velocity projection. 3. A simple planar example. The nal geodesic curve is as straight line as expected. (a) Top view (the ^ C curve) of the process. (b) The nal geodesic curve is presented on the 3D surface. 4. Example with an \egg" box surface. The initial curve (^ C 0) is given by a sin function on the (x; y){plane. 5. Example with a Gaussian surface. The initial curve is given by a sin function on the (x; y){plane. 6. Example of two diierent initial curves evolving according to the proposed shortening ow. review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed," J. 13] M. Grayson, \The heat equation shrinks embedded plane curves to round points," J. Acknowledgments We wish to thank Prof. James A. Sethian for being always so open and generous in providing us his works. Especially, we thank him for the reports 6, 7, 25], which are closely connected to the work here described. Appendix A The exact numerical equation corresponding to (2) and (10) is now described. If the curve ^ C is a level set of a function , then its curvature ^ can be computed as 24]: ^ = yy 2 x ? 22 x y xy + xx 2 y ((2 x + 2 y) 3=2 ; and the surface evolution equation corresponding to (10) is given by