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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">{{citation 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 ] ]. This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref>{{citation
| last = Aicardi | first = F.
| editor-last = Arnol'd | editor-first = V. I.
| contribution = Tree-like curves
| isbn = 0-8218-0237-2
| location = Providence, Rhode Island
| mr = 1310594
| pages = 1–31
| publisher = American Mathematical Society
| series = Advances in Soviet Mathematics
| title = Singularities and bifurcations
| volume = 21
| year = 1994}}</ref><ref name="Shapiro-1997">{{citation
| last = Shapiro | first = Boris | last = Shapiro | first = Boris
| editor-last = Tabachnikov | editor-first = S. | editor-last = Tabachnikov | editor-first = S.
Line 16: Line 27:
| title = Differential and symplectic topology of knots and curves | title = Differential and symplectic topology of knots and curves
| volume = 190 | volume = 190
| year = 1999}}</ref>
| 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==

Revision as of 08:49, 13 January 2025

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. 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
  2. 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

See also

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