Curve shortening and its application to multi-agent systems

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05

Volumn 2005, Issue , 2005, Pages 2817-2822

  Smith, Stephen L ^a   Broucke, Mireille E ^a   Francis, Bruce A ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL GEOMETRY; LINEAR SYSTEMS; PROBLEM SOLVING; THEOREM PROVING; VECTORS;

EMBEDDED CURVE; EUCLIDEAN CURVE; GAGE-HAMILTON-GRAYSON THEOREM; POLYGON SHORTENING;

MULTI AGENT SYSTEMS;

EID: 33847205357     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/CDC.2005.1582590     Document Type: Conference Paper

References (22)

1. S. Angenent, G. Sapiro, and A. Tannenbaum. On the affine heat equation for nonconvex curves. J. of the American Math. Soc., 11:601-634, 1998.
2. D. A. Brannan, M. F. Esplen, and J. J. Gray. Geometry. Cambridge University Press, 1999.
3. A. M. Bruckstein, G. Sapiro, and D. Shaked. Evolution of planar polygons. International Journal of Pattern Recognition and Artificial Intelligence, 9(6):991-1014, 1995.
4. E. Calabi, P. Olver, and A. Tannenbaum. Affine geometry, curve flows, and invariant numerical approximations. Advances in Mathematics, 124:154-196, 1996.
5. F. Cao. Geometric Curve Evolution and Image Processing. Springer, 2003.
6. K-S. Chou and X-P. Zhu. The Curve Shortening Problem. Chapman and Hall, 2001.
7. H. S. M. Coxeter. Regular Polytopes. Dover Publications, Inc., New York, third edition, 1973.
8. P. J. Davis. Circulant Matrices. Chelsea Publishing, New York, 2nd edition, 1979.
9. M. E. Gage. An isoperimetric inequality with applications to curve shortening. Duke Mathematical Journal, 50(3): 1225-1229, 1983.
10. M. E. Gage. Curve shortening makes convex curves circular. Inventiones mathematicae, 76:357-364, 1984.
11. M. E. Gage and R. S. Hamilton. The heat equation shrinking convex plane curves. J. Differential Geometry, 23:69-96, 1986.
12. M. A. Grayson. The heat equation shrinks embedded plane curves to round points. J. Differential Geometry, 26:285-314, 1987.
13. M. A. Grayson. Shortening embedded curves. Annals of Mathematics, 129:71-111, 1989.
14. H. W. Guggenheimer. Differential Geometry. Dover Publications, Inc., New York, 1977.
15. T. Jecko and J-C. Leger. Polygon shortening makes (most) quadrilaterals circular. Bull. Korean Math. Society, 39(1):97-111, 2002.
16. Z. Lin, M. Broucke, and B. Francis. Local control strategies for groups of mobile autonomous agents. IEEE Trans. on Automatic Control, 49(4):622-629, 2004.
17. M. S. Mel'nikov. Analytic capacity: discrete approach and curvature of measure. Sbornik: Math., 186(6):827-846, 1995.
18. K. Nakayama, H. Segur, and M. Wadati. A discrete curve-shortening equation. Methods and Applications of Analysis, 4(2): 162-172, 1997.
19. R. Osserman. Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly, 86:1-29, 1979.
20. H. G. Rotstein, S. Brandon, and A. Novick-Cohen. Hyperbolic flow by mean curvature. Journal of Crystal Growth, 198/199:1256-1261, 1999.
21. G. Sapiro and A. Tannenbaum. On affine plane curve evolution. Journal of Functional Analysis, 119:79-120, 1994.
22. S. L. Smith, M. E. Broucke, and B. A. Francis. A hierarchical cyclic pursuit scheme for vehicle networks. Automatica, 41(6):1045-1053, 2005.