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{{short description|Type of planar curve with tree-like structure}} {{short description|Type of planar curve with tree-like structure}}
] ]
In ], particularly in ], a '''tree-like curve''' is a ] ] <math>c: S^1 \to \mathbb{R}^2</math> with the property that removing any ] splits the curve into exactly two ] ].<ref name="Shapiro-1997">Shapiro, B. (1997). "Tree-like curves and their number of inflection points". arXiv:dg-ga/9708009</ref> This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref name="Shapiro-1997"/> In ], particularly in ], a '''tree-like curve''' is a ] ] <math>c: S^1 \to \mathbb{R}^2</math> with the property that removing any ] splits the curve into exactly two ] ].<ref name="Shapiro-1997">{{citation
| last = Shapiro | first = Boris
| editor-last = Tabachnikov | editor-first = S.
| arxiv = dg-ga/9708009
| contribution = Tree-like curves and their number of inflection points
| doi = 10.1090/trans2/190/08
| isbn = 0-8218-1354-4
| location = Providence, Rhode Island
| mr = 1738394
| pages = 113–129
| publisher = American Mathematical Society
| series = American Mathematical Society Translations, Series 2
| title = Differential and symplectic topology of knots and curves
| volume = 190
| year = 1999}}</ref> This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref name="Shapiro-1997"/>


==References== ==References==

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Type of planar curve with tree-like structure
A tree-like curve with finitely many marked double points

In mathematics, particularly in differential geometry, a tree-like curve is a generic immersion c : S 1 → R 2 {\displaystyle 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.

References

  1. ^ 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

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