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Mémoire sur la représentation des transcendantes par des arcs de courbes

Allégret (1873)

Annales scientifiques de l'École Normale Supérieure

Mémoire sur les systèmes généraux de courbes planes algébriques ou transcendantes, définis par deux caractéristiques

Fouret (1873/1874)

Bulletin de la Société Mathématique de France

Minimal enclosing hyperbolas of line sets.

Schröcker, Hans-Peter (2007)

Beiträge zur Algebra und Geometrie

Modeling repulsive forces on fibres via knot energies

Simon Blatt, Philipp Reiter (2014)

Molecular Based Mathematical Biology

Modeling of repulsive forces is essential to the understanding of certain bio-physical processes, especially for the motion of DNA molecules. These kinds of phenomena seem to be driven by some sort of “energy” which especially prevents the molecules from strongly bending and forming self-intersections. Inspired by a physical toy model, numerous functionals have been defined during the past twenty-five years that aim at modeling self-avoidance. The general idea is to produce “detangled” curves having...

Modules pour les familles de courbes planes

Jean-Paul Dufour (1989)

Annales de l'institut Fourier

L’étude des familles de courbes plane différentiables se ramène a celle des diagrammes $ℝ \overset{f}{\leftarrow} S \overset{σ}{\to} ℝ^{2}$ où $S$ est une surface, $f$ et $σ$ étant différentiables. Dans la classification de ces diagrammes à équivalence près il apparaît trois types de modules: des modules locaux attachés à chaque fronce de $σ$ , des modules semi-locaux attachés à la superposition en un même point de plusieurs situations locales, des modules globaux attachés aux “courbes de contact” le long desquelles certaines courbes sont tangentes. Nous explicitons...

More on the Determinacy of Smooth Map-Germs.

Terence Gaffney, A. de Plessis (1982)

Inventiones mathematicae

Motion by curvature of planar networks

Carlo Mantegazza, Matteo Novaga, Vincenzo Maria Tortorelli (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two–dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which...