Distance-Induced Vector Fields Associated to Triangles

• Luiz Fernando Gonçalves ^[1] ; Ronaldo Alves Garcia ^[1]
  1. [1] Universidade Federal de Goiás

     Universidade Federal de Goiás

     Brasil

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Given three distinct non-collinear points in the Euclidean plane, we study the locally Lipschitz planar vector field associated with the gradient of the sum of the distances to these points.We characterize its equilibrium points in terms of the geometry of the triangle generated by the vertices. In particular, when all interior angles are smaller than 120◦, there exists a unique equilibrium point that coincides with the classical Fermat point of the triangle.We investigate the qualitative properties of the associated gradient flow, proving the existence of a Lyapunov structure and describing the global dynamics of the system. The singular behavior at the vertices is analyzed through a polar blow-up, revealing a saddle-node transition in the local dynamics when the critical angle 120◦ is crossed. We also study the orthogonal dynamics associated with the distance-induced vector field, obtaining a complementary Hamiltonian-type structure tangent to the level sets of the distance function. Near the vertices, the blowup of the orthogonal dynamics exhibits corner-like configurations. These level sets correspond to generalized multifocal ellipses, relating the dynamics to the classical geometry of multifocal conics.

• Referencias bibliográficas
  • 1. Kuhn, H.W.: A note on Fermat’s Problem. Math. Program. 4(1), 98–107 (1973). https://doi.org/10. 1007/BF01584648
  • 2. Spindler, K.: The Fermat-Torricelli problem revisited.Math. Semesterber. 73(1), 51–76 (2026). https:// doi.org/10.1007/s00591-025-00402-y
  • 3. Weber, A.: Theory of the Location of Industries. Hassell Street Press, Kutztown, USA (2021)
  • 4. Morrison, K.E.: The FedEx Problem. Coll. Math. J. 41(3), 222–232 (2010). https://doi.org/10.4169/ 074683410X488719
  • 5. Plastria, F.: Four-point Fermat location problems revisited. New proofs and extensions of old results. IMA J. Manag. Math. 17(4), 387–396...
  • 6. Mello, L.F., Santos, L.: On the location of the minimum point in the euclidean distance sum problem. São Paulo J. Math. Sci. 12(1), 108–120...
  • 7. Sekino, J.: n-ellipses and minimum distance sum problem. Am. Math. Mon. 106(3), 193–202 (1999). https://doi.org/10.1080/00029890.1999.12005031
  • 8. Weiszfeld, E., Plastria, F.:On the point forwhich the sum of the distances to n given points isminimum. Ann. Oper. Res. 167(1), 7–41 (2009)....
  • 9. Groß, C., Strempel, T.-K.: On generalizations of conics and on a generalization of the Fermat-Torricelli Problem. Am. Math. Mon. 105(8),...
  • 10. Sotomayor, J.: Lições de Equações Diferenciais Ordinárias, segunda Projeto Euclides. IMPA, Rio de Janeiro (2025)
  • 11. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems, Unabridged, corrected republication of the 1976 english original, published...
  • 12. Dumortier, F., Llibre, J., Artés, J.: Qualitative Theory of Planar Differential Systems. Universitext, p. 302. Springer, Berlin (2006)....
  • 13. Roussarie, R.: Bifurcation of PlanarVector Fields and Hilbert’s Sixteenth Problem.Modern Birkhäuser Classics, vol. 164, p. 206. Birkhäuser,...
  • 14. Erd˝os, P., Vincze, I.: On the approximation of convex, closed plane curves by multifocal ellipses. Essays in statistical science, Pap....
  • 15. Melzak, Z.A., Forsyth, J.S.: Polyconics. I: polyellipses and optimization. Q. Appl. Math. 35, 239–255 (1977). https://doi.org/10.1090/qam/448883
  • 16. Varga, A., Vincze, C.: On a lower and upper bound for the curvature of ellipses with more than two foci. Expo. Math. 26(1), 55–77 (2008)....
  • 17. Gutierrez, C.: A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 12(6),...
  • 18. Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited. New Mathematical Library, vol. 19, p. 193. Random House, Inc., New York, New York...
  • 19. Tabachnikov, S.: Fagnano orbits of polygonal dual billiards. Geom. Dedicata. 77(3), 279–286 (1999). https://doi.org/10.1023/A:1005199315944
  • 20. Troubetzkoy, S.: Dual billiards, Fagnano orbits, and regular polygons. Amer. Math. Monthly 116(3), 251–260 (2009). https://doi.org/10.4169/193009709X470092