Tree-like curve: Difference between revisions

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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 ] ]. This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref>{{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.		\| last = Aicardi \| first = F.
	\| editor-last = Arnol'd \| editor-first = V. I.		\| editor-last = Arnol'd \| editor-first = V. I.

Revision as of 19:16, 13 January 2025

Type of planar curve with tree-like structure

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.

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

Revision as of 08:49, 13 January 2025 editDavid Eppstein (talk \| contribs)Autopatrolled, Administrators226,556 edits 2nd source← Previous edit		Revision as of 19:16, 13 January 2025 edit undoDavid Eppstein (talk \| contribs)Autopatrolled, Administrators226,556 edits dabsNext edit →
Line 1:		Line 1:
	{{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 ] ]. This property gives these curves a ]-like structure, hence their name. They were first systematically studied by ] ] ] and ] in the 1990s.<ref>{{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.		\| last = Aicardi \| first = F.
	\| editor-last = Arnol'd \| editor-first = V. I.		\| editor-last = Arnol'd \| editor-first = V. I.