5.3 Projective Invariant Polygonal Flow

each. The one after 5 iterations and the other after 100 iterations. The two sets of polygons are projective transformations of each other. Figure 13: Two projective transformed initial polygons evolve identically, at 5 and 100 iterations. 6 Concluding Remarks We have attempted to outline a rather comprehensive theory of iterative invariant smoothings for planar curves and polygons, generalizing and extending some previous results. Many interesting questions remain open and we expect to address these in the near future. Acknowledgment: We thank Guillermo Sapiro for many interesting and useful discussions on invariant evolutions. Thanks are also due to Sanjeev Kulkarni and Tom Richardson for interesting discussions, and to Jee Lagarias for pointing out to AMB the reference 19]. The other implementation problem is that, as with all other polygonal evolutions, we cannot let an iteration take a vertex to the corresponding invariant average. Doing this would cause severe instability eeects. Rather we would like to move the vertex gradually towards the invariant average. The implementation of this gradual progress is multiplying the diierence vector by a damping factor , as in Figure 12a. The problem in this implementation is that it is not projective invariant. To overcome this problem we may use yet another projective invariant, the cross ratio. A D B C α Figure 12: The second problem in projective invariant ow: Making the ow smooth. The cross ratio of four collinear points A, B, C, and D, is a ratio of line-segment lengths as follows CR(A; B; C; D) = jACj jBDj jADj jBCj The cross ratio is projective invariant. If we know three points A, C, and D, and decide about a certain cross ratio, we can deduce a projective invariant B. If the cross ratio is close to 1, then B is close to A. The three invariant points we use to deduce the new vertex in the projective invariant iterative scheme, are the source vertex, the destination, (i.e. the invariant average) and another point, the rst intersection between the line going from the source to the destination and a line from the convex hull of the four nearest vertices, see Figure 12b. The above described algorithm was implemented. In addition to the above mentioned implementation problems, the algorithm suuers from numerical problems so that usually we could not make too many iterations. Therefore, we can not draw general conclusions about the limiting shape of the …