Tree-like curve

In mathematics, particularly in differential geometry, a tree-like curve is a generic immersion $c:S^{1}\to \mathbb {R} ^{2}$ with the property that removing any double point splits the curve into exactly two disjoint connected components. This property gives these curves a tree-like structure, hence their name. They were first systematically studied by Russian mathematicians Boris Shapiro and Vladimir Arnold in the 1990s.^[1]^[2]

References

^ Aicardi, F. (1994), "Tree-like curves", in Arnol'd, V. I. (ed.), Singularities and bifurcations, Advances in Soviet Mathematics, vol. 21, Providence, Rhode Island: American Mathematical Society, pp. 1–31, ISBN 0-8218-0237-2, MR 1310594
^ Shapiro, Boris (1999), "Tree-like curves and their number of inflection points", in Tabachnikov, S. (ed.), Differential and symplectic topology of knots and curves, American Mathematical Society Translations, Series 2, vol. 190, Providence, Rhode Island: American Mathematical Society, pp. 113–129, arXiv:dg-ga/9708009, doi:10.1090/trans2/190/08, ISBN 0-8218-1354-4, MR 1738394